
Glass _i2iZ3 



Book 



MUi 



A 

TEXT BOOK 



OF 



GEOMETRICAL DRAWING, 

^irdrfielr from tj)e ©ctabo Hiritfon, 



FOR THE 

USE OF SCHOOLS, 

IN WHICH 

THE DEFINITIONS AND RULES OF GEOMETRY ARE FAMILIARLY EXPLAINED, THE 

PRACTICAL PROBLEMS ARE ARRANGED FROM THE MOST SIMPLE TO THE 

MORE COMPLEX, AND IN THEIR DESCRIPTION TECHNICALITIES 

ARE AVOIDED AS MUCH AS POSSIBLE; 

WITH AN 

INTRODUCTION TO ISOMETRICAL DRAWING, 

AND AN 

ESSAY ON LINEAR PERSPECTIVE AND SHADOWS: 

THE WHOLE ILLUSTRATED WITH 

FORTY-EIGHT STEEL PLATES. 
BY WILLIAM MINIFIE, 

ARCHITECT, AND TEACHER OP DRAWING IN THE CENTRAL HIGH SCHOOL 
OF BALTIMORE. 



PUBLISHED BY WM. MINIFIE & CO., 

NO. 114 BALTIMORE STREET, 
BALTIMORE. 

1849. 



Entered according to Act of Congress, in the year 1849, 

BY WILLIAM MTNIFIE, 
in the Clerk's Office of the District Court of Maryland. 



STEREOTYPED AT THE » » 

» • • 

BALTIMORE TYPE AND STEREOTYPE ' rOUNDRY, 
FIEJiDING LUCAS, JR., PROPRIETOR. 



^^ 



\^ 



fc' 



ADVERTISEMENT 



THIS 

ABRIDGMENT 

OF 

M I N I Fl E'S TEXT BOOK 

OP 

GEOMETRICAL DRAWING 

Has been made to meet the views of many 
Teachers^ as well as of members of School 
Committees^ who were desirous of introducing 
the work into the Institutions under their 
charge^ provided its bulk and cost could be 
reduced. 

The favorable reception and extensive sale 
of the octavo edition^ has encouraged us to en- 
deavor to meet their wishes^ by the publication 
of the present volume. It is reduced to a con- 
venient size for a school book^ and the uncom- 
monly low price at which it is issued will 
doubtless induce its extensive adoption. 



PREFACE 



Having been for several years engaged in teaching Architec- 
tural and Mechanical Drawing, both in the High School of Bal- 
timore and to private classes, I have endeavored without success, 
to procure a book that I could introduce as a text book; works 
on Geometry generally contain too much theory for the purpose, 
with an insufficient amount of practical problems; and books on 
Architecture and Machinery are mostly too voluminous and cost- 
ly, containing much that is entirely unnecessary for the purpose. 
Under these circumstances, I collected most of the useful practi- 
cal problems in geometry from a variety of sources, simplified 
them and drew them on cards for the use of the classes, arrang- 
ing them from the most easy to the more difficuh, thus leading 
the students gradually forward; this was followed by the draw- 
ing of plans, sections, elevations and details of Buildings and 
Machmery, then followed Isometrical Drawing, and the course 
was closed by the study of Linear Perspective and Shadows ; the 
whole being illustrated by a series of short lectures to the private 
classes. 

I have been so well pleased with the results of this method of 
instruction, that I have endeavored to adopt its general features 
in the arrangement of the following work. The problems in 
constructive geometry have been selected with a view to their 
practical application in the every-day business of the Engineer, 
Architect and Artizan, while at the same time they afford a good 
series of lessons to facilitate the knowledge and use of the instru- 
ments required in mechanical drawing. 

The definitions and explanations have been given in as plain 
and simple language as the subject will admit of; many persons 
will no doubt think them too simple. Had the book been intend- 
ed for the use of persons versed in geometry, very many of the 
explanations might have been dispensed with, but it is intended 
chiefly to be used as a first book in geometrical drawing, by per- 
sons who have not had the benefit of a mathematical education, 
and who in a majority of cases, have not the time or inclination 



VI 



PREFACE. 



to Study any complex matter, or what is the same thing, that 
which may appear so to them. And if used in schools, its de- 
tailed explanations, we believe, will save time to the teacher, by 
permitting the scholar to obtain for himself much information 
that he would otherwise require to have explained to him. 

But it is also intended to be used for self -instruction, without 
the aid of a teacher, to whom the student might refer for expla- 
nation of any difficulty ; under these circumstances I do not be- 
lieve an explanation can be couched in too simple language. 
With a view of adapting the book to this class of students, the 
illustrations of each branch treated of, have been made progres- 
sive, commencing with the plainest diagrams ; and even in the 
more advanced, the object has been to instil principles rather than 
to produce effect, as those once obtained, the student can either 
design for himself or copy from any subject at hand. It is hoped 
that this arrangement will induce many to study drawing who 
would not otherwise have attempted it, and thereby render them- 
selves much more capable of conducting any business, for it has 
been truly said by an eminent writer on Architecture, " that one 
workman is superior to another (other circumstances being the 
same) directly in proportion to his knowledge of drawing, and 
those who are ignorant of it must in many respects be subser- 
vient to others who have obtained that knowledge." 

The size of the work has imperceptibly increased far beyond 
my original design, which was to get it up in a cheap form with 
illustrations on wood, and to contain about two-thirds of the 
number in the present volume, but on examining some speci- 
mens of mathematical diagrams executed on wood, I was dis- 
satisfied with their want of neatness, particularly as but few stu- 
dents aim to excel their copy. On determining to use steel illus- 
trations, I deemed it advisable to extend its scope, until it has at- 
tamed its present bulk, and even now I feel more disposed to 
increase than to curtail it, as it contains but few examples either 
in Architecture or Machinery. I trust, however, that the objec- 
tor to its size will find it to contain but little that is absolutely 
useless to a student. 

In conclusion, I must warn my readers against an idea that I 
am sorry to find too prevalent, viz : that drawing requires but 
little time or study for its attainment, that it may be imbibed in- 
voluntarily as one would fragrance in a flower garden, with little 
or no exertion on the part of the recipient, not that the idea is 



PREFACE. 



Vll 



expressed in so many words, but it is frequently manifested by 
their dissatisfaction at not being able to make a drawing in a few 
lessons as well as their teacher, even before they have had suffi- 
cient practice to have obtained a free use of the instruments. I 
have known many give up the study in consequence, who at 
the same time if they should be apprenticed to a carpenter, would 
be satisfied if they could use the jack plane with facility after 
several weeks practice, or be able to make a sash at the end of 
some years. 

Now this idea is fallacious, and calculated to do much injury ; 
proficiency in no art can be obtained without attentive study and 
industrious perseverance. Drawing is certainly not an excep- 
tion ; but the difficulties will soon vanish if you commence with 
a determination to succeed ; let your motto be persevere, never 
say " it is too difficult ;" you will not find it so difficult as you 
imagine if you will only give it proper attention ; and if my 
labors have helped to smooth those difficulties it will be to me a 
source of much gratification. 



WM. MINIFIE. 



Baltimore, 1st March 1849. 



ILLUSTRATIONS. 


' 




PLATE. 


Definitions of lines and angles, .... 


i. 


Definitions of plane rectilinear superficies, . 


ii. 


Definitions of the circle, 


iii. 


To erect or let fall a perpendicular, .... 


iv. 


Construction and division of angles. 


V. 


Construction of polygons, 


vi. 


Problems relating to the circle, .... 


. vii. 


Parallel ruler, and its application, . . . 


viii. 


Scale of chords and plane scales, .... 


ix. 


Protractor, its construction and application, . 


x. 


Flat segments of circles and parabolas. 


xi. 


Oval figures composed of arcs of circles. 


xii. 


Cycloid and Epicycloid, 


. xiii. 


Cube, its sections and surface, 


xiv. 


Prisms, square pyramid and their coverings, . 


. XV. 


Pyramid, Cylinder, Cone and their surfaces. 


xvi. 


Sphere and covering and coverings of the regular Poly he 


- 


drons, 


\ xvii. 


Cylinder and its sections, . . . 


xviii. 


Cone and its sections, 


. xix. 


Ellipsis and Hyperbola, . . . . . 


XX. 


Parabola and its application to Gothic arches, . 


. xxi. 


Coverings of hemispherical domes, . . . 


xxii. 


Joints in Circular and Elliptic arches. 


xxiii. 


Joints in Gothic arches, . . . 


xxiv. 


Octagonal plan and elevation, .... 


. XXV. 


Circular plan and elevation, ... 


xxvi. 


Roman mouldings, 


xxvii. 


Grecian mouldings, 


xxviii. 


To proportion the teeth of wheels, .... 


. xxix. 


Isometrical cube, its construction, . • . . 


xxx. 


Isometrical figures, triangle and square, . 


. xxxi. 



X 


ILLUSTRATIONS. 








PLATE. 


Iso metrical figures pierced and chamfered, . 


xxxii* 


Isometrical 


circle, method of describing and dividing it. 


xxxiii. 


Perspective- 


—Visual angle, section of the eye, Stc. 


xxxiv. 


<( 


Foreshortening and definitions of lines. 


XXXV. 


(( 


Squares, half distance, and plan of a room. 


xxxvi. 


a 


Tessellated pavements. 


xxxvii. 


<£ 


Square viewed diagonally. Circle. . 


xxxviii. 


iS 


Line of elevation, pillars and pyramids. 


xxxix. 


({ 


Arches parallel to the plane of the picture. 


xl. 


iS 


Arches on a vanishing plane. 


. xli. 


(( 


Application of the circle. 


xlii. 


i( 


Perspective plane and vanishing points. 


. xliii. 


C( 


Cube viewed accidentally. 


xliv. 


(e 


Cottage viewed accidentally. 


. xlv. 


t( 


Frontispiece. Street parallel to the middle 






visual ray, .... 


. xlvi. 


Shadows, rectangular and circular, .... 


xlvii. 


Shadows of 


steps and cylinder, .... 


xlviii. 



DEFINITIONS OFZINESJiND JINGLES 




M 



H. 




l71;'yuui.ScSo:^ 



PRACTICAL GEOMETRY. 



PLATE I. 

DEFINITIONS OF LINES AND ANGLES, 



1. A Point is said to have position without magni- 
tude ; and it is therefore generally represented to the 
eye by a small dot, as at A. 

2. A Line is considered as length without breadth or 
thickness, it is in fact a succession of points ; its ex- 
tremities therefore, are points. Lines are of three 
kinds ; right lineSj curved lines^ and mixed lines. 

3. A Right Line, or as it is more commonly called, 
a straight line^ is the shortest that can be drawn between 
two given points as JS. 

4. A Curve or Curved Line is that which does not 
lie evenly between its terminating points, and of 
which no portion, however small, is strai^'ht; it is 
therefore longer than a straight line connecting the 
same points. Curved lines are either regular or irre- 
gular. 

5. A Regular Curved Line, as (7, is a portion of 
the circumference of a circle, the degree of curvature 
being the same throughout its entire length. An ir- 
regular curved line has not the same degree of curva- 
ture throughout, but varies at different points. 

6. A Waved Line may be either regular or irregu- 
lar; it is composed of curves bent in contrary directions. 



12 PLATE I. 

£ is a regular waved line, the inflections on either 
side of the dotted line being equal; a waved line is 
also called a line of double curvature of contrary flex- 
ure^ and a serpentine line. 

7. Mixed Lines are composed of straight and curv- 
ed lines, as D. 

8. Parallel Lines are those which have no incli- 
nation to each other, as F^ being every where equidis- 
tant; consequently they could never meet, though pro- 
duced to infinity. 

If the parallel lines G were produced, they would 
form two concentric circles, viz : circles which have a 
common centre, whose boundaries are every where par- 
allel and equidistant. 

9. Inclined Lines, as H and I, if produced, would 
meet in a point as at if, forming an angle of which 
the point K is called the vertex or angular point, and 
the lines H and / the legs or sides of the angle K; 
the point of meeting is also called the summit of an 
angle. 

10. Perpendicular Lines. — Lines are perpendic- 
ular to each other when the angles on either side of 
the point of junction are equal ; thus the lines JV. 0. P 
are perpendicular to the line L. M, The lines Jf. 0. P 
are called also vertical lines and plumb lines, because 
they are parallel with any line to which a plummet is 
suspended ; the line L, JIf is a horizontal or level line; 
lines so called are always perpendicular to a plumb line. 

11. Vertical and Horizontal Lines are always 
perpendicular to each other, but perpendicular lines are 
not always vertical and horizontal ; they may be at any 
inclination to the horizon provided that the angles on 
either side of the point of intersection are equal, as for 
example the lines X F and Z. 

12. Angles. — Two right lines drawn from the same 



FUU Z 



BEEimrmNs. piAm bectiuneab svperficies 



TBTAXGLBS OB. TRfGOys . 




QVJiBRILATEllALS,QTJADBAN&LES OR TETRAGOlffS 
PAMALLELOItRMIS , 



W 



^0 



BECTANGLES 



r^j 






m 



1=^- 




MTTUin 



PLATE II, 



13 



point, diverging from each other, form an angle, as the 
lines S. Q. R, An angle is commonly designated by- 
three letters, and the letter designating the point of di- 
vergence, which in this case is Q, is always placed in 
the middle. Angles are either acute, right or obtuse. 
If the legs of an angle are perpendicular to each other, 
they form a right angle as T, Q. i?, (mechanics' 
squares, if true, are always right angled;) if the sides 
are nearer together, as *S. Q. i?, they form an acute 
angle; if the sides are wider apart, or diverge from 
each other more than a right angle, they form an obtuse 
angle, as V, Q. jR. 

The magnitude of an angle does not depend on the 
length of the sides, but upon their divergence from each 
other; an angle is said to be greater or less than an- 
other as the divergence is greater or less; thus the ob- 
tuse angle F. Q. R is greater, and the acute angle S, 
Q, R is less than the right angle T, Q. R. 



PLATE IL 

PLANE RECTILINEAR SUPERFICIES, 



13. A Superficies or Surface is considered as an 
extension of length and breadth without thickness. 

14. A Plane Superficies is an enclosed flat sur- 
face that will coincide in every place with a straight 
line. It is a succession of straight lines, or to be more 
explicit, if a perfectly straight edged ruler be placed on 
a plane superficies in any direction, it would touch it 
in every part of its entire length. 

15. When surfaces are bounded by right lines, they 



14 PLATE II. 

are said to be Rectilinear or Rectilineal. As all 
the figures on plate second agree with the above defi- 
nitions, they are Plane Rectilinear Superficies. 

16. Figures bounded by more than four right lines 
are called Polygons ; the boundary of a polygon is 
called its Perimeter. 

17. When Surfaces are bounded by three right 
lines, they are called Triangles or Trigons. 

18. An Equilateral Triangle has all its sides of 
equal length, and all its angles equal, as A. 

19. An Isosceles Triangle has two of its sides 
and two of its angles equal, as B, 

20. A Scalene Triangle has all its sides and 
angles unequal, as C 

21. An Acute Angled Triangle has all its angles 
acute, as A and B. 

22. A Right Angled Triangle has one right 
angle; the side opposite the right angle is called the 
hypothenuse ; the other sides are called respectively the 
base and perpendicular. The figures A. B. C, are 
each divided into two right angled triangles by the 
dotted lines running across them. 

23. An Obtuse Angled Triangle has one obtuse 
angle, as C 

24. If figures A and B were cut out and folded on 
the dotted line in the centre of each, the opposite sides 
would exactly coincide; they are therefore, regular 
triangles. 

25. Any of the sides of an equilateral or scalene tri- 
angle may be called its Base, but in the Isosceles tri- 
angle the side which is unequal is so called, the angle 
opposite the base is called the Vertex. 

26. The Altitude of a Triangle is the length of a 
perpendicular let fall from its vertex to its base, as a. 
A, and h, B^ or to its base extended, as rf. d, figure C. 



PLATE II. 



15 



The superficial contents of a Triangle may be obtain- 
ed by multiplying the altitude by one half the base. 

27. When surfaces are bounded by four right lines, 
they are called Quadrilaterals, Quadrangles or 
Tetragons ; either of the figures D. E, F. G. H and 
K may be called by either of those terms, which are 
common to all four-sided right lined figures, although 
each has its own proper name. 

28. When a Quadrilateral has its opposite sides par- 
allel to each other, it is called a Parallelogram ; 
therefore figures D. E. F and G are parallelograms. 

29. When all the angles of a Tetragon are right an- 
gles, the figure is called a Rectangle, as figures D 
and E. 

If two opposite angles of a Tetragon are right angles, 
the others are necessarily right too. 

30. If the sides of a Rectangle are all of equal 
length, the figure is called a Square, as figure D. 

31. If the sides of a Rectangle are not all of equal 
length, two of its sides being longer than the others, as 
figure Eyit is called an Oblong. 

32. When the sides of a parallelogram are all equal, 
and the angles not right angles, two being acute and 
the others obtuse, as figure Fj it is called a Rhomb, or 
Rhombus ; it is also called a Diamond, and sometimes 
a Lozenge, more particularly so when the figure is 
used in heraldry. 

33. When the sides of a parallelogram are not all 
equal, and the angles not right angles, but whose oppo- 
site sides are equal, as figure G, it is called a Rhom- 
boid. 

34. If two of the sides of a Quadrilateral are parallel 
to each other as the sides H and in fig. if, it is call- 
ed a Trapezoid. 

35. All other Quadrangles are called Trapeziums, 



16 



PLATE II. 



the terra being applied to all Tetragons that have no 
two sides parallel, as K. 

Note.— The terms Trapezoid and Trapezium are appUed 
indiscriminately by some writers to either of the figures if and K; 
by others^ fig. H is called a Trapezium and fig. K a Trapezoid, 
and this appears to be the more correct method; but as Trapezoid 
is a word of comparatively modern origin, I have used it as it is 
most generally applied by modern writers, more particularly so 
in works on Architecture and Mechanics. 

36. A Diagonal is a line running across a Quad- 
rangle, connecting its opposite corners, as the dotted 
lines in figs. D and F. 

Note. — I have often seen persons much confused, in conse- 
quence of the number of names given to the same figure, as for 
example, fig. D. 

1st. It is Biplane Figure — see paragraph 14* 

2nd. It is Rectilineal, being composed ol right lines* 

3rd* It is a Quadrilateral, being bounded by four lines. 

4th. It is a Quadrangle, having four angles. 

5th. It is a Tetragon, having four sides. 

6th* It is a Parallelogram, its opposite sides being parallel. 

7th. It is a Rectangle, all its angles being right angles. 

All the above may be called common names, because they are 
applied to all figures having the same properties. 

8th. It is a Square, which is its prober name, distinguishing it 
from all other figures, to which some or all of the above terms 
may be applied. 

All of them except 7 and 8, may also be applied to fig. jP, with 
the same propriety as to fig. D; besides these, fig. F has four 
proper names distinguishing it from all other figures, viz: a 
Rhomb, Rhomhus, Diamond and Lozenge. 

If the student will analyze all the other figures in the same 
manner, he will soon become perfectly familiar with them, and 
each term will convey to his mind a clear definite idea. 




Plate. S. 



DEFINITIONS OF THE CIRCLE 



SEMICmCLE . 



n A 



SEGMENTS 




7. 
SUPPLEMENT . 




10. 

co-snsfE 



<^^^^^^m 



o . 
quAniiANr. 



COMPLEMENT. 
E 




'Yj^\ 



TANGENT Sec 




11 



H 



n Ar 






9. 
SINE 




r^ 






?r"^Minifi&. 



PJmMriy Sc Sens 



PLATE III. 



17 



PLATE III. 

DEFINITIONS OF THE CIRCLE. 



1. A Circle is a plane figure bounded by one 
curve line, every where equidistant from its centre, as 
fig. 1. 

2. The boundary line is called the Circumfeu- 
ENCE or Periphery, it is also for convenience called a 
Circle. 

3. The Centre of a circle is a point within the 
circumference, equally distant from every point in it, as 
C,fig. 1. 

4. The Radius of a circle is a line drawn from the 
centre to any point in the circumference, as C. Ji^ C B 
or C D, fig. 1. The plural of Radius is Radii. All 
radii of the same circle are of equal length. 

5. The Diameter of a circle is any right line 
drawn through the centre to opposite points of the cir- 
cumference, as ^. jB, fig. 1. The length of the di- 
ameter is equal to two radii ; there may be an infinite 
number of diameters in the same circle, but they are all 
equal. 

6. A Semicircle is the half of a circle, as fig. 2; 
it is bounded by half the circumference and by a di- 
ameter. 

7. A Segment of a circle is any part of the surface 
cut off by a right line, as in fig. 3. Segments may be 
therefore greater or less than a semicircle. 

8. An Arc of a circle is any portion of the circum- 
ference cut off, as C. G. D or E. G. F, fig. 3. 



18 PLATE III. 

9. A Chord is a right line joining the extremities 
of an arc, as C. D and E, F^ fig. 3. The diameter is 
the chord of a semicircle. The chord is also called 
the Subtense. 

10. A Sector of a circle is a space contained be- 
tween two radii and the arc which they intercept, as E, 
a D, or 0. a H, fig. 4. 

11. A Quadrant is a sector whose area is equal 
to one- fourth of the circle, as fig. 5 ; the arc D. E be- 
ing equal to one-fourth of the whole circumference, and 
the radii at right angles to each other. 

12. A Degree. — The circumference of a circle is 
considered as divided into 360 equal parts called De- 
grees, (marked^) each degree is divided into 60 
minutes (marked') and each minute into 60 seconds 
(marked''); thus if the circle be large or small, the 
number of divisions is always the same, a degree being 
equal to l-360th part of the whole circumference, the 
semicircle equal to 180°, and the quadrant equal to 
90°. The radii drawn from the centre of a circle to 
the extremities of a quadrant are always at right angles 
to each other ; a right angle is therefore called an angle 
of 90°. If we bisect a right angle by a right line, it 
would divide the arc of the quadrant also into two 
equal parts, each part equal to one-eighth of the whole 
circumference containing 45° ; if the right angle were 
divided into three equal parts by straight lines, it would 
divide the arc into three equal parts, each containing 
30°. Thus the degrees of the circle are used to mea- 
sure angles, and when we speak of an angle of any 
number of degrees, it is understood, that if a circle 
with any length of radius^ be struck with one foot of the 
dividers in the angular pointy the sides of the angle will 
intercept a portion of the circle equal to the number of 
degrees given. 



PLATE III. 



19 



Note. — ^This division of the circle is purely arbitrary, but it 
has existed from the most ancient times and every where. Dur- 
ing the revolutionary period of 1789 in France, it was proposed 
to adopt a decimal division, by which the circumference was 
reckoned at 400 grades; but this method was never extensively 
adopted and is now virtually abandoned. 

13. The Complement of an Jlrc or of an Angle is 
the difference between that arc or angle and a quad- 
rant; thus E. D fig. 6 is the complement of the arc D. 
By and E. C, D the complement of the angle D. C B. 

14. The Supplement of an Arc or of an Angle is 
the difference between that arc or angle and a semicir- 
cle; thus D, A fig. 7, is the supplement of the arc D. 
By and D. C, A the supplement of the angle JB. C. D. 

15. A Tangent is a right line, drawn without a cir- 
cle touching it only at one point as B. E fig. 8; the 
point where it touches the circle is called the point of 
contact, or the tangent point. 

16. A Secant is a right line drawn from the centre 
of a circle cutting its circumference and prolonged to 
meet a tangent as C. E fig. 8. 

Note. — Secant Point is the same as point of intersection, 
being the point where two lines cross or cut each other. 

17. The Co-Tangent of an arc is the tangent of the 
complement of that arc, as H, K fig. 8. 

Note. — The shaded parts in these diagrams are the given angles^ 
but if in fig. 8, D. C. H be the given angle and D. H the given 
arc, then H. K, would be the tangent and B. E the co-tangent. 

18. The Sine of an arc is a line drawn from one 
extremity, perpendicular to a radius drawn to the other 
extremity of the arc as D. F fig. 9. 

19. The Co-sine of an arc is the sine of the comple- 
ment of that arc as L, D fig. 10. 

20. The Versed Sine of an arc is that part of the 
radius intercepted between the sine and the circumfer- 
ence as F. B fig. 9. 



20 



PLATE III, 



21. In figure 11, we have the whole of the foregoing 
definitions illustrated in one diagram. C. H — C. D — 
C B and C. A are Radii; A. B the Diameter ; B, C. D 
a Sector ; jB. C il a Quadrant, Let B. C. D be the 
given Angle J and B, D the given Arc^ then JB. D. is 
the Chord ^ D. H the Complement^ and J). ^ the 5^p- 
plement of the arc ; D. C, H. the Complement and D. 
C .y? the Supplement of the given angle ; B. E the 
Tangent and If. iC the Co-tangentj C. E the Secant 
and C ii the Co-secant ^ F. D the S'zne, i. D the Cb- 



sine and JP. jB the Versed Sine. 



PLATE IV. 
TO ERECT OR LET FALL A PERPENDICULAR. 

Problem 1. Figure 1. 



To bisect the right line A. B by a perpendicular, 

1st. With any radius greater than one half of the 
given line, and with one point of the dividers in A and 
B successively, draw two arcs intersecting each other, 
in C and D. 

2nd. Through the points of intersection draw C. D, 
which is the perpendicular required. 



Problem 2. Fig. 2, 



From the point D in the line E. F to erect a petyen- 

dicular. 

1st. With one foot of the dividers placed in the given 



Plate. 4 . 


TO EIXECT OR LET FALL A FERPEWDlCULAIt . 




&'. 


y 






:^ 1 


Fuj.l . 


Tf 


Fig.Z. 






F, •' 






B D r 


F 


D; 


II 




F 

( 


\ 


Ftc).3. / 






/ . H 


\ ty-4. 




/''' 'G 


/ 




c s^ -, ^ H B B r 


D 


p ^ Fu^.ii. ; 




-Fl^ , 5 . 


'\,3<\ 




, \E 


F/ „ . -fi.--' ' ■ F 


/' ^ 1 


. o 




'"~---- ---' ^ 

A 


E 

s 

A 


c; 


«( 


:d 






''\ '■■■■., Fiff.8. /"' 




F^^.Z 






^ E F 








G - - y / 




;K 






c 15 ;i 


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c\l_'.. ■-■''' 



Vr'M-iruJic . 



Bbrwiu &:So?is. 



PLATE IV. 



21 



point D with any radius less than one half of the line, 
describe an arc, cutting the given line in jB and C 

2nd. From the points B and C with any radius 
greater than J5. D, describe two arcs, cutting each 
other in G. 

3rd. From the point of intersection draw G. J), 
which is the perpendicular required. 



Problem 3. Fig. 3. 



To erect a perpendicular when the point D is at or near 
the end of a line, 

1st. With one foot of the dividers in the given point 
D, with any radius, as D. E^ draw an indefinite arc G. H, 

2nd. With the same radius and the dividers in any 
point of the arc, as E^ draw the arc B. jD. jP, cutting 
the line C. D in B. 

3rd. From the point B through E draw a right line, 
cutting the arc in F, 

4th. From F draw F. D, which is the perpendicular 
required. 

Note. — It will be perceived that the arc B. D. F is a semicir- 
cle, and the right Hne B. F a diameter ; if from the extremities 
of a semicircle right lines be drawn to any point in the curve, the 
angle formed by them will be a right angle. This affords a ready 
method for forming a ^'square corner,^' and will be found useful 
on many occasions, as its accuracy may be depended on. 



Problem 4. Fig. 4. 



Another method of erecting a perpendicular when at or 
near the end of the line. 

Continue the line H. D toward C, and proceed as in 
problem 2 ; the letters of reference are the same. 



22 



PLATE IV. 



Problem 5. Fig. 5. 



From the point D to let fall a perpendicular to the line 

A. B. 

1st. With any radius greater than D. G and one foot 
of the compasses in D, describe an arc cutting A. B in 
E and F. 

2nd. From E and F with any radius greater than E. 
G, describe two arcs cutting each other as in C. 

3rd. From D draw the right line D. C, then D. G is 
the perpendicular required. 

Problem 6. Fig. 6. 



When the point D is nearly opposite the end of the line, 

1st. From the given point D, draw a right line to 
any point of the line A. B as 0. 

2nd. Bisect 0. D by problem 1, in JEf. 

3rd. With one foot of the compasses in E with a ra- 
dius equal to E, D ot E, describe an arc cutting A. 
BinF. 

4th. Draw D. Fy which is the perpendicular required. 

Note. — The reader will perceive that we have arrived at the 
same result as we did by problem 3, but by a different process, 
the right angle being formed within a semicircle. 



Problem 7. Fig. 7. 



Another method of letting fall a perpendicular when the 
given point D is nearly opposite the end of the line. 

1st. With any radius as jP. D and one foot of the 
compasses in the line A. B as at F^ draw an arc D. 

H. a 



HMMH 



PLATE IV. 



23 



2nd. With any other radius as E. D draw another 
arc D. K C, cutting the first arc in C and D. 

3rd. From D draw D. C, then D. G is the perpen- 
dicular required. 

Note. — The points E and F from which the arcs are drawn, 
should be as far apart as the line A. B will admit of, as the exact 
points of intersection can be more easily found, for it is evident, 
that the nearer two lines cross each other at a right angle, the 
finer will be the point of contact. 



Problem 8. Fig. 8. 



To erect a perpendicular at D the end of the line C. D, 
with a scale of equal parts. 

1st. From any scale of equal parts take three in your 
dividers, and with one foot in D, cut the line C. D in B. 

2nd. From the same scale take four parts in your 
dividers, and with one foot in D draw an indefinite arc 
toward E. 

3rd. With a radius equal to five of the same parts, 
and one foot of the dividers in B^ cut the other arc in E. 

4th. From E draw E. D, which is the perpendicular 
required. 

Note 1st. If four parts were first taken in the dividers and laid 
off on the fine C, D, then three parts should be used for striking 
the indefinite arc, at A, and the five parts struck from the point 
C, which would give the intersection A, and arrive at the same 
result. 

2nd. On referring to the definitions of angles, it will be found 
that the side of a right angled triangle opposite the right angle is 
called the Hypothenuse; thus the line E. B is the hypothenuse 
of the triangle E. D. B, 

3rd. The square of the hypothenuse of a right angled triangle 
is equal to the sum of the squares of both the other sides. 

4th. The square of a number is the product of that number 
multiplied by itself. 



24 PLATE IV. 

Example. — ^The length of the side D. E is 4, which muhiplied 
by 4, will give for its square 16. The length of D. B is 3, which 
multiplied by 3, gives for the square 9. The products of the two 
sides added together give 25. The length of the hypothenuse is 
5, which multiplied by 5, gives also 25. 

5th. The results will always be the same, but if fractional parts 
are used in the measures, the proof is not so obvious, as the mul- 
tiplication would be more comphcated. 

6th. 3, 4 and 5 are the least whole numbers that can be used 
in laying down this diagram, but any multiple of these numbers 
may be used -, thus, if we multiply them by 2, it would give 6, 8 
and IO3 if by 3, it would give 9, 12 and 15; if by 4—12, 16 and 
20, and so on. The greater the distances employed, other things 
being equal, the greater will be the probable accuracy of the result. 

7th. We have used a scale of equal parts without designating 
the unit of measurement, which may be an inch, foot, yard, or 
any other measure. 

8th. As this problem is frequently used by practical men in 
laying off work, we will give an illustration. 

Example. — Suppose the line C. D to be the front of a house, 
and it is desired to lay off the side at right angles to it from the 
corner D, 

1st. Drive in a small stake at D, put the ring of a tape mea- 
sure on it and lay off twelve feet toward JB. 

2nd. With a distance of sixteen feet, the ring remaining at D, 
trace a short circle on the ground at E. 

3rd. Remove the ring to jB, and with a distance of twenty feet 
cut the first circle at E. 

4th. Stretch a line from D to E, which will give the required 
side of the building. 



FJate. 5. 



ro]vsTRurTU>]\r.iJVB onnsiox of jxcles . 




Fi/j . 2 . 





%• ^• 




Fi^.5. ^. 





'" JflfZl/lf 



^^nuvt 2s. ^ V. 



PLATE V* 



25 



PLATE V. 

CONSTRUCTION AND DIVISION OF ANGLES. 

Problem 9. Fig, 1. 



The length of the sides of a Triarigle A. B., C. D. and 
E. F being given^ to construct the Triangle, the two 
longest sides to be joined together at A. 

1st. With the length of the line C, D for a radius 
and one foot in Ay draw an arc at G. 

2nd. With the length of the line E. F for a radius 
and one foot in J5, draw an arc cutting the other arc 
at G. 

3rd. From the point of intersection draw G. A and 
G. By which complete the figure. 



Problem 10. Fig. 2. 
To construct an Angle at K equal to the Angle H, 

1st. From Hwith any radius, draw an arc cutting 
the sides of the angle as at M. JY. 

2nd. From K with the same radius, describe an in- 
definite arc at 0. 

3rd. Draw K parallel to H. M. 

4th. Take the distance from JIf to JV^ and apply it 
from to P. 

5th. Through P draw K P, which completes the fi- 
gure. 



26 






PLATE 


V. 




Problem 11. Fig. 


3. 

a Right 


Line. 




To Bisect the given Angle Q by 





1st. With any radius and one foot of the dividers in 
Q draw an arc cutting the sides of the angle as in R 
and S, 

2nd. With the same or any other radius, greater than 
one-half JR. 5', from the points S and i?, describe two 
arcs cutting each other, as at T, 

3rd. Draw T. Q, which divides the angle equally. 

Note. — This problem may be very usefully applied by workmen 
on many occasions. Suppose^ for example^, the corner Q be the 
corner of a room, and a washboard or cornice has to be fitted 
around it; first, apply the bevel to the angle and lay it down on 
a piece of board, bisect the angle as above, then set the bevel to 
the centre line, and you have the exact angle for cutting the mitre. 
This rule will apply equally to the internal or external angle. 
Most good practical workmen have several means for getting the 
cut of the mitre, and to them this demonstration will appear un- 
necessary, but I have seen many men make sad blunders, for 
want of knowing this simple rule. 



Problem 12. Fig. 4, 



To Trisect a Right Angle. 

1st. From the angular point Fwith any radius, de- 
scribe an arc cutting the sides of the angle, as in X 
and W. 

2nd. With the same radius from the points X and 
Wj cut the arc in Y and Z. 

3rd. Draw Y. V and Z. F, which will divide the 
angle as required. 



PLATE V, 



27 



Problem 13. Fig. 5. 



To inscribe in the triangle A. B. Cj a Circle touching all 

its sides, 

1st. Bisect two of the angles by problem 11, as •/? 
and JS, the dividing lines will cut each other in D, then 
D is the centre of the circle. 

2nd. From B let fall a perpendicular to either of the 
sides as at F^ then D, F is the radius, with which to 
describe the circle from the point B, 

Problem 14. Fig. 6. 



On the given line A. B ^o construct an Equilateral 
Triangle J the line A . ^ to he one of its sides, 

1st. With a radius equal to the given line from the 
points ^ and By describe two arcs intersecting each 
other in C. 

2nd. From C, draw C, A and C B^ to complete the 
figure. 



I 

CO 
A polygon 


PLATE VI. 
NSTRUCTION OF POLYGONS. 


of 3 sides is called \ 

4 u u 

5 " " 

6 '' " 

7 u u 

8 '' 

9 '' " 


a Trigon. 
Tetragon. 
Pentagon. 
Hexagon. 
Heptagon. 
Octagon. 
EnneagonorNonagon. 



28 



PLATE VI. 



A polygon of 10 sides is called a Decagon. 
" 11 '^ '' Undecagon. 

" 12 '' "- Dodecagon. 

1st. When the sides of a polygon are all of equal 
length and all the angles are equal, it is called a regu- 
lar polygon ; if unequal, it is called an irregular polygon. 

2nd. It is not necessary to say a regular Hexagon, 
regular Octagon, &c. ; as when either of those figures 
is named, it is always supposed to be regular, unless 
otherwise stated. 



Problem 15. Fig. 1 



On a given line A. B i^o construct a square ivhose side 
shall be equal to the given line. 

1st. With the length^. B for a radius from the 
points A and J5, describe two arcs cutting each other 

in a 

2nd. Bisect the arc C. A or C. B in D. 

3rd. From C, with a radius equal to C D, cut the 
arc B, E in E and the arc A. F in jP. 

4th. Draw A, E, E. F and F. B, which complete 
the square. 

Problem 16. Fig. 2. 



In the given square G. H. K. J, to inscribe an Octagon. 

1st. Draw the diagonals G. K and H. J, intersecting 
each other in P. 

2nd. With a radius equal to half the diagonal from 
the corners G. H. K and J, draw arcs cutting the sides 
of the square in 0. 0. 0, &c. 

3rd. Draw the right lines 0. 0., 0. 0, &c., and 
they will complete the octagon. 



Phrtv 0'. 



CONSTlUrCTlON OF VOLVaoNS 



rig. I. 

E ,F 



(' 



\r> 



Fuj. ?. . 



r/ 



() u 



e: 





" D 




PLATE VI. 



29 



This mode is used by workmen when they desire to 
make a piece of wood round for a roller, or any other 
purpose; it is first made square, and the diagonals 
drawn across the end ; the distance of one half the di- 
agonal is then set off, as from G to jR in the diagram, 
and a guage set from H to R which run on all the cor- 
ners, gives the lines for reducing the square to an octa- 
gon ; the corners are again taken off, and finally finish- 
ed with a tool appropriate to the purpose. The centre 
of each face of the octagon gives a line in the circum- 
ference of the circle, running the whole length of the 
piece ; and as there are eight of those lines equidistant 
from each other, the further steps in the process are 
rendered very simple. 



Problem 17. Fig. 3. 



In a given circle to inscribe an Equilateral Triangle^ a 
Hexagon and Dodecagon, 

1st. For the Triangle, with the radius of the given 
circle from any point in the circumference, as at Jl, de- 
scribe an arc cutting the circle in jB and C. 

2nd. Draw the right line B. C, and with a radius 
equal B, C, from the points B and C, cut the circle 
in.D. 

3rd. Draw D. B and D. C, which complete the tri- 
angle. 

4th. For the Hexagon, take the radius of the given 
circle and carry it round on the circumference six times, 
it will give the points .yi. B, E, D. F, C, through them, 
draw the sides of the hexagon. The radius of a circle 
is always equal to the side of an hexagon inscribed 
therein. 



30 PLATE VI. 

5th. For the Dodecagon, bisect the arcs between 
the points found for the hexagon, which will give the 
points for inscribing the dodecagon. 



Problem 18. Fig. 4. 



hi a given Circle to inscribe a Square and an Octagon, 

1st. Draw a diameter Jl. B, and bisect it with a per- 
pendicular by problem 1, giving the points C D. 

2nd. From the points A. C. B, D, draw the right 
lines forming the sides of the square required. 

3rd. For the Octagon, bisect the sides of the square 
and draw perpendiculars to the circle, or bisect the arcs 
between the points A, C, B. D, which will give the 
other angular points of the required octagon. 



Problem 19. Fig. 5\ 



On the given line 0. V to construct a Pent agon , O. P 
being the length of the side. 

1st. With the length of the line 0. P from 0, de- 
scribe the semicircle P. Q, meeting the line P. 0, ex- 
tended in Q. 

2nd. Divide the semicircle into five equal parts and 
from draw lines through the divisions 1, 2 and 3. 

3rd. With the length of the given side from P, cut 
1 in 5', from S cut 2, in P, and from Q cut 2 
in R; connect the points O. Q. P. S, P by right lines, 
and the pentagon will be complete. 



PLATE VI. 



31 



Problem 20. Fig. 6. 



On the given line A. B to construct a Heptagon^ A. B 
being the length of the side, 

1st. From A with A, B for a radius, draw the semi- 
circle B, H. 

2d. Divide the semicircle into seven equal parts, and 
from A through 1, 2, 3, 4 and 5, draw indefinite lines. 

3rd. From B cut the line ^ 1 in C, from C cut A 2 
in D, from G cut A 4 in jP, and from F cut A 3 in jE, 
connect the points by right lines to complete the figure. 

Any polygon may be constructed by this method. 
The rule is, to divide the semicircle into as many equal 
parts as there are sides in the required polygon, draw 
lines through all the divisions except two, and proceed 
as above. 

Considerable care is required to draw these figures 
accurately, on account of the difficulty of finding the 
exact points of intersection. They should be practised 
on a much larger scale. 



PLATE VII 



Problem 21. Fig. 1. 



To find the Centre of a Circle. 

1st. Draw any chord, as A, U, and bisect it by a 
perpendicular E. D, which is a diameter of the circle. 

2nd. Bisect the perpendicular £. D by problem 1, 
the point of intersection at C is the centre of the circle. 



32 PLATE VII. 



Figure 2, 



Another method of Jiiiding the Centre of a Circle, 

1st. Join any three points in the circumference as 
F. G. H. 

2nd. Bisect the chords F. G and G. H by perpen- 
diculars, their point of intersection at C is the centre 
required. 



Problem 22. Fig. 3. 



To draw a Circle through any three points not in a 
straight line^ as M. N. 0. 

1st. Connect the points by straight lines, which will 
be chords to the required circle. 

2nd. Bisect the chords by perpendiculars, their point 
of intersection at C is the centre of the required circle. 

3rd. With one foot of the dividers at C, and a radius 
equal to C JVJ, C. JV, or C 0, describe the circle. 



Problem 23. Fig. 4. 



To find the Centre for describing the Segment of a 

Circle, 

1st. Let P. R be the chord of the segment, and P. 
>S the rise; 

2nd. Draw the chords P. Q and Q. R, and bisect 
them by perpendiculars; the point of intersection at C 
is the centre for describing the segment. 



FUde 7. 



VllOBLEMS RELATING TO THE CIRCLE, 






Fuf. 4 %: 



\ Fig.S 




PLATE VII, 



33 



Problem 24 Fig. 5. 



To find a Right Line nearly equal to an Arc of a 
Circle^ a5 H. I. K. 

1st. Draw the chord H. K, and extend it indefinitely 
toward O. 

2nd. Bisect the segment in I, and draw the chords 
H. I and /. K. 

3rd. With one foot of the dividers in i?, and a radius 
equal to H, I, cut H. O in Jkf, then with the same ra- 
dius, and one foot in Jkf, cut if. again in JY. 

4th. Divide the diiference K. JY into three equal 
parts, and extend one of them toward O, then will the 
right line H. be nearly equal to the curved line 
H. I. K. 



Problem 25. Fig. 6< 



To find a Right Line nearly equal to the Semicircum" 
ference A. F. B. 

1st. Draw the diameter^. jB, and bisect it by the 
perpendicular F, H; extend F. H indefinitely toward G. 

2nd. Divide the radius C H into four equal parts, 
and extend three of those parts to G. 

3rd. At F draw an indefinite right line D. E, 

4th. From G through A^ the end of the diameter A. 
jB, draw G. A, D, cutting the line D. E in D, and from 
G through B draw G. B, E, cutting D. E in J5, then 
will the line D. E be nearly equal to the semicircum- 
ference of the circle, and the triangles D. G. E and A. 
G. B will be equilateral. 

Note. — ^The right lines found by problems 24 and 25, are not 
mathematically equal to the respective curves, but are sufficiently 



34 PLATE VII. 

correct for all practical purposes. Workmen are in the habit of 
using the following method for finding the length of a curved 
line: — 

They open their compasses to a small distance^ and commenc- 
ing at one end, step oJET the whole curve, noting the number of 
steps required, and the remainder less than a step, if any ; they 
then step off the same number of times, with the same distance, 
on the article to be bent around it, and add the remainder, which 
gives them a length sufficiently true for their purpose : the error 
in this method amounts to the sum of the differences between the 
arc cut off by each step, and its chord. 



PLATE VIII. 
PARALLEL RULER AND APPLICATION. 

Figure 1. 



The parallel ruler figured in the "plate, consists of 
two bars of ^yood or metal A,. B and C D, of equal 
length, breadth and thickness, connected together by 
two arms of equal length placed diagonally across the 
bars, both at the same angle, and moving freely on the 
rivets which connect them to the bars; if the bar.y3. B 
be kept firmly in any position and the bar C D moved, 
the ends of the arms connected to CV D will describe 
arcs of circles and recede from A. B until the arms are 
at right angles to the bars, as shewn by the dotted 
lines; if moved farther round, the bars will again ap- 
proach each other on the other side. 

The bars of which the ruler is composed, being par- 
allel to each other, it follows, that if either edge of the 
instrument is placed parallel with a line and held in 
that position, another line may be drawn parallel to the 
first at any distance within the range of the instrument. 



TARALLEL RULER 

and its AppUmUorb 



Fig.l 




'W'^MuufLf.. 



Wntxi/i 1:Sl 



PLATE ViU. 35 

This is its most obvious use; it is generally applied to 
the drawing of inclined parallel lines in mechanical 
drawings, vertical and horizontal lines being more 
easily drawn with the square, when the drawing is at- 
tached to a drawin'g board. 

Application. — Problem 26. Fig. 2. 



To divide the Line E. F into any nicmher of equal 
parts y say 12. 

1st. From E draw E, G at any angle to E. F, and 
step off with any opening of the dividers, twelve equal 
spaces on E. G, 

2nd. Join jP 12, and w^ith the parallel ruler draw 
lines through the points of division in E. G, parallel to 
12 F intersecting E. Fj and dividing it as required. 

Problem 27. Fig. 3. 



To divide a Line of the length of G. H in the same fro- 
portion as th^ Line I. K is divided, 

1st. From / draw a line at any angle and make /, L 
equal to G. H, 

2nd. Join the ends K and L by a right line, and 
draw lines parallel to it through all the points of divi- 
sion, to intersect J. i, then J. L will be divided in the 
same proportion as /. K. 

Problem 28. Fig. 4. 



To reduce the Trapezium A. B. C. D to a Triangle of 

equal area, 

1st. Prolong C, D indefinitely. 



r 



36 PLATE VIII. 

2nd. Draw the diagonal ji. D, place one edge of the 
ruler on the line ./I. D and extend the other edge to By 
then draw B, E, cutting C. D extended in E, 

3rd. Join A. Ey then the area of the triangle A, E, 
C will be equal to the trapezium A. J5. C D. 



Problem 29. Fig. 5. 



To reduce the irregular Pentagon F. G. H. L K. to a 
Tetragon and to a Triangle ^ each of equal area with 
the Pentagon. 

1st. Prolong I. 11 indefinitely. 

2nd. Draw the diagonal F, H and G. M parallel to 
it, cutting /. H in M, and draw F, M, 

3rd. Prolong K, I indefinitely toward L. 

4th. Draw the diagonal F, I and draw M, L parallel 
to it, cutting /i. L in i. 

5th. Draw F, X, then the triangle F, L, K, and the 
tetragon K. F. M, /, are equal in area to the given 
pentagon. 



PLATE IX 



CONSTRUCTION OF THE SCALE OF CHORDS 
AND ITS APPLICATION— PLANi^SCALES. 

Problem 30. Figure 1. 



To Construct a Scale of Chords, 

Let *y2. B be a rule on which to construct the scale. 
1st. With any radius, and one foot in .D, describe a 
quadrant; then draw the radii D. Cand D. E, 



PLATE IX. 37 

2nd. Divide the arc into three equal parts as fol- 
lows : — With the radius of the quadrant, and the divi- 
ders in C, cut the arc in 60 ; then, with one foot in E, 
cut the arc in 30. 

3rd. Divide these spaces each into three equal parts, 
when the quadrant will be divided into rune equal parts 
of 10^ each. 

4th. From C, draw chords to each of the divisions, 
and transfer them, as shewn by the dotted lines, to Jl, B. 

5th. Divide each of the spaces on the arc into ten 
equal parts, and transfer the chords to A. By when we 
shall have a scale of chords corresponding to the re- 
spective degrees. 

Note 1. — ^This scale is generally found on the plane scale 
which accompanies a set of drawing instruments, and marked C, 
or Ch. 

Note 2. — Any scale of chords may be re-constructed by using 
the chord of 60^ as a radius for describing the quadrant. 



Application. — -Problem 31. Fig. 2. 



To lay down an Angle at F, of any number of degrees ^ 
say 25, the line G. F to form one side of the Angle, 

1st. Take the chord of 60^ in the dividers, and with 
one foot in jP, describe an indefinite arc, cutting G. F 
in H. 

2nd. From the scale take 25^ in the dividers, and 
with one foot in i/, cut the arc in K. 

3rd. Through Jf, draw IL F, which completes the 
required angle. If we desire an angle of 15^ or 30°, 
take the required number from the scale, and cut the 
arc in or P, and in the same manner for any other 
anirle. 



38 PLATE IX. 



Problem 32. Fig. 2. 



To measure art Jingle F, already laid down. 

1st. With the chord of 60^, and one foot of the di- 
viders in the angular point, cut the sides of the angle 
in H and K. 

2nd. Take the distance H. K in the dividers, and 
apply it to the Scale, which will shew the number of 
degrees subtended by the angle. 



SCALES OF EQUAL PARTS. 



1st. Scales of equal parts may be divided into two 
kinds, viz : Thbse which consist of two or three lines, 
divided by short parallel lines, at right angles to the 
others, like fig. 3, or those which are composed of sev- 
eral parallel lines, divided by diagonal and vertical 
lines, like figs. 4 and 5 : the first kind are called simple 
scales^ the second diagonal scales. 

2nd. Scales of equal parts may be made of any size, 
and may be made to represent any unit of measure : 
thus each part of a scale may be an inch, or the tenth 
of an inch, or any other space, and may represent an 
inch, foot, yard, fathom, mile, or degree, or any other 
quantity. 

3rd. The measure which the scale is intended to 
represent, is called the unit of measurement. In archi- 
tectural or mechanical drawings, the unit of measure- 
ment is generally a foot, which is subdivided into 
inches to correspond with the common foot rule. For 
working drawings, the scale is generally large. A very 
common mode of laying down working drawings, is to 



Platp.f) . 
SCJLLE OF rHORDS 




n~ III I I I I ~~T~ I llllllllll r 1 

\o s\o 7\o e\o 5\o A\i) 3\o 2\o i\o I R 




SCALES OF EQUAL PAJXTS 

Fig, 3 . 

lO 9 8 765432. 1 O 



m m I I I 



H 



I I 



O I 2 



4 5 6 7 



L5 



Fig. 4 



A. 





1 


2 


3 


4 


5 


6 


7 


A. 


















/ 


Y 


















L 


\. 


















L \; 


















\ 







































Fiif . 5 




10 987 6S432 



PLATE IX. 39 

use a scale of one and a half inches to the foot; this 
gives one-eighth of an inch to an inch, which is equal 
to one-eighth of the full size. This is very convenient, 
as every workman has a scale on his rule, which he 
can apply to the drawing with facility. Scales are 
generally made to suit each particular case, dependant 
on the size of the object to be represented, and on the 
size of the paper or board on which the drawing is to 
be made. 

4th. To DRAW A Scale. If we have a definite size 
for each part of a required scale, say one-quarter of an 
inch, we have only to extend the dividers to that mea- 
sure, step off the parts and number them, reserving the 
left hand space to be subdivided for inches. Care 
should be taken to have the scale true. It should be 
proved by taking two, three, four or five parts in the 
dividers, and applying it to several parts of the scale ; 
when, if found correct, the drawing may be proceeded 
with. It is much easier to draw another scale if the 
first is imperfect, than to correct a drawing made from 
a false scale. 

5th. Fig. 3 requires but little explanation. It is 
called a quarter of an inch scale, as each unit of the 
scale is a quarter of an inch; the starting point of a 
scale marked 0, is called its Zero. The term is not 
very common among practical men, except when ap- 
plied to the thermometer; and, when we say the ther- 
mometer is down to Zero, we mean that it is at the 
commencement of the scale. It is better to number a 
scale above and below, as in the figure ; for, if we wish 
to take a measure of any number of feet less than 10, 
and inches, say 3 feet 6 inches, we place one foot of 
the dividers at 3, numbered from below, and extend the 
other out to 6 inches. If, on the other hand, we have 
to measure a number of feet more than 10, say 13, we 



40 PLATE IX. 

should place one foot in the division marked 10, on 
the top of the scale without the plate, and extend it to 
3, when we are enabled to read the quantity at sight, 
without any mental operation, as we must do if the 
scale is only numbered as below. For example, to 
take 13 feet, we must place one foot of the dividers at 
20, and extend it to 7. This operation is simple, it is 
true, but it requires us to subtract 7 from 20 to get 13, 
instead of reading from the scale as we could do from 
the upper numbers. In taking a large space in the di- 
viders, it is always better to take the whole numbers 
first, and add the inches or other fractions afterwards. 
The space on the left hand in this figure, is divided into 
twelve parts for inches. 



Figure 4 

Is A HALF INCH DIAGONAL ScALE, divided for feet 
and inches. To draw a scale of this kind : — 

1st. Draw 7 lines parallel to each other, and equi- 
distant. 

2nd. Step off* spaces of half an inch each, and draw 
lines through the divisions across the whole of the 
spaces. 

3rd. Divide the top of the first space into two equal 
parts, draw the diagonal lines, and number them as in 
the diagram. 

To take any measure from this scale, say 2' 1", we 
must place the dividers on the first line above the bot- 
tom on the second division on the scale, and extend 
the other foot to the first diagonal line, numbered 1, 
which will give the required dimension. If w^e wish 
to take 2^ 11'', we must place one leg of the dividers 
the same as before, and extend the other to the second 



PLATE IX. 41 

diagonal line, which gives the dimension. If we wish 
to take 1' 3^^, we must place the dividers on the middle 
line in the first vertical division, and extend it to the 
first diagonal line, numbered 3, and proceed in the 
same manner fi)r any other dimension. 

This is a very useful form of scale. The student 
should familiarize himself with its construction and ap- 
plication. 



Figure 5 

Is AN INCH Diagonal Scale, divided into tenths 
and hundredths. 

It is made by drawing eleven lines parallel to each 
other, enclosing ten equal spaces, with vertical lines 
drawn through the points of division across the whole. 
The left hand vertical space is divided into ten equal 
parts, and diagonal lines drawn as in the figure. 

This scale gives three denominations. Each of the 
small spaces on the top and bottom lines, is equal to 
one-tenth of the whole division. The horizontal lines 
contained betw^een the first diagonal and the vertical 
line, are divided into tenths of the smaller division, or 
hundredths of the larger division ; for example, the first 
line from the top contains nine-tenths of the smaller 
division, the second eight- tenths, the third seven-tenths, 
and so on as numbered on the end of the scale. To 
make a diagonal scale of this form, divided into feet, 
inches, &c., we must draw 13 parallel horizontal lines, 
and divide the left hand space also into 13. 



42 PLATE X. 



PLATE X. 

CONSTRUCTION OF THE PROTRACTOR. 

Figure 1 



The protractor is an instrument generally formed of 
a semicircle and its chord; the semicircle is divided 
into 180 equal parts or degrees, numbered in both di- 
rections from 10^ to 180^5 as in its application, angles 
are often required to be measured or laid down on 
either hand; in portable cases of instruments the pro- 
tractor is frequently drawn on a flatstraight scale as in 
the diagram. Its mode of construction is sufficiently 
obvious from the drawing; a small notch or mark in 
the centre of the straight edge of the instrument denotes 
the centre from which the semicircle is described, and 
the angular point in which all the lines meet. 



Application. — Problem 33. Fig. 2, 



WITH THE PROTRACTOR, TO PROTRACT OR 
LAY DOWN ANY ANGLE. 

From the point O let it he required to form a Right 
Angle to the line O. P. 

1st. Place the straight edge of the protractor to coin- 
cide with the line 0. P, with the centre at 0, then 
mark the angle of 90*^ at S, 

2nd. From S draw S, O, which gives the required 
angle. 



fbitti 10 . 

mOTRAC'TOR . 

Its i'mistrux'Jioio ami AppUralion 



Fuj.l 




Amjles of Fvegidar Polvij 



ons 



Trill on 

Tl'tfY/{JOJI.... 

Fer I tn (](>!! ... 
W^.riifiini ... 
Ortdficii .... 
Eriiu'dfjcn . 
Dt'r/ifjmi ... 
Do(h'((HN>ii 



OO 
7^ 
GO 
45 
41^ 
3G 
30 



at the' ffjiti'e ! 


ov 


at tlh^ (ircainleTYiuv 


//"...... 


vo 


//'.'.- 


//': 


IZO 


//'.'.... 


.r..J 

1 


fV'..... 


^v: J 


./.v'/T 


....//':..._ 


ji"......: 


14 O 


^"-. 


^r... .] 


144 


. d'\.._. 


- --/':..' 


r. 



hium .1' 



PLATE X, 



43 



While the instrument is in the position described, 
with its centre at O, any other angle may be laid down, 
thus at Q we have 30°, at R 60°, at T 120°, and at 
F150°, and so on from the fraction of a degree up to 
180°. 

The protractor may also be used for constructing any 
regular polygon in a circle or on a given line; to do so, 
it is necessary to know the angle formed by said poly- 
gon by lines drawn from its corners to the centre of the 
circle, and also to know the angle formed by any two 
adjoining faces of the polygon. The table given for 
this purpose is constructed as follows : 

1st. To find the angle formed by any polygon at the 
centre, divide 360, the number of degrees in the whole 
circle, by the number of sides in the required polygon, 
the quotient will be the angle at the centre; for exam- 
ple, let it be required to find the angle at the centre of 
an octagon : — divide 360 by 8 the number of sides, the 
quotient will be 45, which is the angle formed by the 
octagon at the centre. 

2nd. To find the angle formed by two adjoining faces 
of a polygon, we must subtract from 180 the number 
of degrees in the semicircle; the angle formed by said 
polygon at the centre, the remainder will be the angle 
formed at the circumference. For example let us take 
the octagon ; we have found in the last paragraph that 
the angle formed by that figure in the centre is 45° ; 
then if we subtract 45 from 180 it will leave 135, which 
is the angle formed by two adjoining faces of the oc- 
tagon. 



44 PLATE XI. 



PLATE XL 

TO DESCRIBE FLAT SEGMENTS OF CIRCLES 
AND PARABOLAS. 



Very often in practice it would be very inconvenient 
to find the centre for describing aflat segment of a cir- 
cle, in consequence of the rise of an arch being so 
small compared to its span. 



Problem 34. Fig. 1 



To describe a Segment with a Triangle. 

Note. — In all the diagrams in this plate A. B is the span of 
the arch^ A, D the rise, and C the centre of the crown of the arch. 

1st. Make a triangle with its longest side equal to 
the chord or span of the arch and its height equal to 
one-half the rise. 

2nd. Stick a nad at A and at C, place the triangle as 
in the diagram and move it round against the nails to- 
ward jij a pencil kept at the apex of the triangle wuU 
describe one-half of the curve. 

3rd. Stick another nail at B, and with the triangle 
moving against C and By describe the other half of the 
curve. 



fUiU^ 11. 



FLAT SEnMEXTS JJVD FARjiBOLAS 




Fuj.4. 



J. 1 Z 3 4 5 C _;f^^-f f ^ 1^ , E 





1 •13 4 5 6 5 4 3 2 1 



1 2 



Fig. 5. 

4 5 C 



4 3 2 1 



\: 


\____\__i_^— )p 


^ 


"1^ 


^^^zt 


5 / 
3 / 


\ 






1 / 


' /^ 


7 


..i.^ 


i :? 3 ^ 


? .5 ^ 


7 5 4 3 


2 1 





PLATE XI. 



45 



Problem 35. Fig. 2, 



To describe the same Curve with strips of wood^ form- 
ing a Triangle, 

1st. Drive a nail at A and another at B^ place one 
strip against A and bring it up to the centre of the 
crown at C 

2nd. Place another strip against B and crossing the 
first at Cj nail them together at the intersection, and 
nail a brace across to keep them in position. 

3rd. With the pencil at C and the triangle formed by 
the strips kept against A and JB, describe the curve 
from'' C toward A^ and from C toward B, 



Problem 36. Fig. 3. 



To draw a Parabolic Curve by the intersection of lines 
forming Tangents to the Curve, 

1st. Draw C 8 perpendicular to A. JB, and make it 
equal to A. D. 

2nd. Join A, 8 and jB. 8, and divide both lines into 
the same number of equal parts, say 8, number them as 
in the figure, draw 1. 1. — 2. 2. — 3. 3., &c., then these 
lines will be tangents to the curve; trace the curve to 
touch the centre of each of those lines between the 
points of intersection. 



Problem 37. Fig. 4. 



To draw the same Curve by another method. 

1st. Divide A. D and B. J5, into any number of 
equal parts, and C D and C. E into a similar number. 



46 PLATE XI. 

2nd. Draw 1. 1.— 2. 2. &c., parallel to A. D, and 
from the points of division in A. D and B. £, draw- 
lines to C. The points of intersection of the respec- 
tive lines, are points in the curve. 

Note. — The curves found, as in figs. 3 and 4, are quicker at 
the crown than a true circular segment ; but, where the rise of 
the arch is not more than one-tenth of the span, the variation 
cannot be perceived. 



Problem 38. Fig. 5. 



To describe a True Segment of a Circle by Intersections, 

1st. Draw the chords A. C and B- (7, and Jl, and 
B. 0'. perpendicular to them. 

2nd. Prolong D. E in each direction to 0. 0'; di- 
vide 0. C, a O, A, D, A. 6, B. 6, and B. E into the 
same number of equal parts. 

3rd. Join the points 1. 1. — 2. 2. &C.5 in A. B and 
0. 0'. 

4th. From the divisions in A. JD, and J?. £, draw 
lines to C. The points of intersection of these lines 
with the former, are points in the curve. A semicircle 
may be described by this method. 



Plate IZ 



OJ^m. FIGURES COMPOSED OF AMOS OF CIRCLES 



Fig.l. 




/6\ 




Fuj.Z. 


"""^ 


c 


'\^ 


"^ 


^^^^ 




^\!/k 




F 


H~'i/' 


4 3\ Z 1 










/''/ ~ ^> 


4 E 


'V>-^ 






D 


' 


-^-^^"' 



7%.. 5 




M/"' if7«z;7y^. 



RhnjjnkSons. 



I PLATE 



XIL 



47 



PLATE XII. 

TO DESCRIBE OVAL FIGURES COMPOSED OF 
ARCS OF CIRCLES. 



Problem 39. Fig. 1 



The length of the Oval A. B. being given^ to describe an 

Oval upon it, 

1st. Divide A. B the given length, into three equal 
parts, in E and F, 

2nd. With one of those parts for a radius, and the 
compasses in E and F successively, describe two cir- 
cles cutting each other in and 0'. 

3d. From the points of intersection in and 0\ 
draw lines through E and jF, cutting the circles in V, 
F'', and F VJ' 

4th. With one foot of the compasses in 0. and 0' 
successively, and with a radius equal to 0. F^, or OJ 
Vj describe the arcs between V. F,' and VJ^ V'^y to 
complete the figure. 



Problem 40. Fig. 2. 



To describe the Oval^ the length A. B, and breadth C. 
D, being given, 

1st. With half the breadth for a radius, and one foot 
in Fy describe the arc C. J5, cutting Ji, B in E. 

2nd. Divide the difference E. B between the semi- 
axes into three equal parts, and carry one of those divi- 
sions toward 4. 



48 PLATE XII. 

3d. Take the distance B 4, and set off on each side 
of the centre F at H and HJ 

4th. With the radius H. HJ describe from iJand W 
as centres, arcs cutting each other in K and KJ 

5th. From K and KJ through H and HJ draw in- 
definite right lines. 

6th. With the dividers in if, and the radius H, A^ 
describe the curve F. Ji. VJ^ and with the dividers in 
HJ describe the curve VJ B, VJ^' 

7th. From K and KJ with a radius equal to K, C, 
describe the curves V, C, VJ and VJ' D. F/'^ to com- 
plete the figure. 



Problem 41. Fig. 3- 



.brother method for describing the Oval^ the length A. 
B, and breadth C. D, being given. 

1st. Draw C. B, and from 5, wath half the trans- 
verse axis, B. Fy cut jB. C in 0. 

2nd. Bisect J?. by a perpendicular, cutting Jt» B 
in P, and C. D in Q. 

3rd. From F^ set off the distance F. P to 1?, and 
the distance F, Q to 5^. 

4th. From Sy through R and P, and from Q through 
jR, draw indefinite lines. 

5th. From P and iJ, and from S and Q, describe 

the arcs, completing the figure as in the preceding 

problem. 

Note. — In all these diagrams, the result is nearly the same. 
Figs. 1 and 2 are similar figures, although each is produced by a 
different process. The proportions of an oval, drawn as figure 
1, must always be the same as in the diagram; but, in figs. 2 
and 3, the proportions may be varied ; but, when the difference 
in the length of the axes, exceeds one- third of the longer one, the 
curves have a very unsightly appearance, as the change of cur- 



Flat& 13 



CYCLOID ANB EPICYCLOID 



Fi^.l. 




y/^Mimfie. 



lUjii^cUvkSons. 



PLATE XIII. 49 

vature is too abrupt. These figures are often improperly called 
ellipses, and sometimes false ellipses. Ovals are frequently used 
for bridges. When the arch is flat, the curve is described from 
more than two centres, but it is never so graceful as the true 
ellipsis. 



PLATE XIII. 

TO DESCRIBE THE CYCLOID AND EPICYCLOID. 

The Cycloid is a curve formed by a point in the cir- 
cumference of a circle, revolving on a level line; this 
curve is described by any point in the wheel of a car- 
riage when rolling on the ground. 

Problem 42. Fig. 1. 



To find any number of Points in the Cycloid Curve by 
the intersection of lines. 

1st. Let G. H be the edge of a straight ruler, and 
C the centre of the generating circle. 

2nd. Through C draw the diameter A, B perpen- 
dicular to G. H, and E. F parallel to G. H; then A. B 
is the height of the curve ^ and E. F t^ the place of the 
centre of the generating circle at every point of its pro- 
gress, 

3rd. Divide the semicircumference from B io A into 
any number of equal parts, say 8, and from A draw 
chords to the points of division. 

4th. From G, with a space in the dividers equal to 
one of the divisions on the circle, step off on each side 
the same number of spaces as the semicircumference is 
divided into, and through the points draw perpendicu- 
lars to G. H : number them as in the dinofram. 



50 



PLATE XIII, 



5th. From the points of division in JE. F^ with the 
radius of the generating circle, describe indelinite arcs 
as shewn by the dotted lines. 

6th. Take the chord A lin the dividers, and with 
the foot at 1 and 1 on the line G. H^ cut the indefinite 
arcs described from 1 and 1 respectively at D and ly, 
then D and D' are points in the curve, 

7th. With the chord A 2, from 2 and 2 in G, H, cut 
the indefinite arcs in / and J' with the chord A 3, 
from 3 and 3, cut the arcs in if and K' and apply the 
other chords in the same manner, cutting the arcs in 
i. Jkf, &c. 

8th. Through the points so found trace the curve. 

Note. — -Each of the indefinite arcs in the diagram represents 
the circle at that point of its revohition, and the points D. J. K, 
&c., the position of the generating point B at each place. This 
curve is frequently used for the arches of bridges, its proportions 
are always constant, viz: the span is equal to the circumference 
of the generating circle and the rise equal to its diameter. Cy- 
cloidal arches are frequently constructed which are not true cy- 
cloids, but approach that curve in a greater or less degree. 



Figure 2. — The Epicycloid. 



This curve is formed by the revolution of a circle 
around a circle, either within or without its circumfer- 
ence, and described by a point B in the circumference 
of the revolving circle. P is the centre of the revolv- 
ing circle, and Q of the stationary circle. 



Problem 43. 



To find Points in the Curve, 
1st. Draw the diameter 8. 8, and from Q the centre, 
draw Q. B at right angles to 8. 8, 



PLATE XIII. 



51 



2nd. With the distance Q. P from Q, describe an 
arc 0. representing the position of the centre P 
throughout its entire progress. 

3rd. Divide the semicircle B, D and the quadrants 
D. 8 into the same number of equal parts, draw chords 
from D to 1, 2, 3, &c., and from Q draw lines through 
the divisions in D. 8 to intersect the curve 0. in 1, 
2, 3, &c, 

4th. With the radius of P from 1, 2, 3, &c., in 0. 
describe indefinite arcs, apply the chords D 1, jD 2, 
&c., from 1, 2, 3, &c., in the circumference of Q, cut- 
ting the indefinite arcs in ^. C E. F, &c., which are 
points in the curve. 



DEFINITIONS OF SOLIDS 



On refering back to our definitions, we find that a 
point has position without magnitude. 

A Line has length, without breadth or thickness, 
consequently has but one dimension. 

A Surface has length and breadth, without thickness, 
consequently has two dimensions, which, multiplied 
together, give the content of its surface. 

A Solid has length, breadth and thickness. These 
three dimensions multiplied together, give its solid con- 
tent. 

Lineal Measure, is the measure of lines. 

Superficial or Square Measure , the measure of 
surfaces. 

Cubic Measure, is the measure of solids. 

For Example. — If we take a cube whose edge mea- 
sures two feet, then two feet is the lineal dimension of 
that line. If the edge is two feet long, the adjoining 



52 PLATE XIV. 

edge is also two feet long, then, two feet multiplied by 
two, gives four feet, which is the superficial content of 
a face of the cube. 

Then, if we multiply the square or superficial con- 
tent, by two feet, which is the thickness of the cube, it 
will give eight feet, which is its solid content. 

Then, two lineal feet is the length of the edge. 
'' io\xY square '' the surface of one side. 

And eight cubic '' the solid content of the cube. 



PLATE XIV. 



THE CUBE OR HEXAHEDRON, ITS SECTIONS 
AND SURFACE. 



Figure 1 



1st. The cube is one of the regular polyhedrons, 
composed of six regular square faces, and bounded by 
twelve lines of equal length ; the opposite sides are all 
parallel to each other. 

2nd. If a cube be cut through tw^o of its opposite 
edges, and the diagonals of the faces connecting them, 
the section will be an oblong rectangular parallelogram, 
as fig. 2. 

3rd. If a cube be cut through the diagonals of three 
adjoining faces, as in fig. 3, the section will be an equi- 
lateral triangle, whose side is equal to the diagonal of 
a face of the cube. Two such sections may be made 
in a cube by cutting it again through the other three 
diagonals, and the second section will be parallel to the 
first. 

4th. If a cube be cut by a plane passing through all 
its sides, the line of section, in each face, to be parallel 



Flate. 14- . 

THE CUBE 

its Sections and Sur/a/:s 




lw.2. 








Vir^MwMe . 



ninuuh kSorhs. 



PLATE XIV. 



53 



with the diagonal, and midway between the diagonal 
and the corner of the face, as in fig. 4, the section will 
be a regular hexagon^ and will be parallel with, and 
exactly midway between the triangular sections de- 
fined in the last paragraph. 

5th. If a cube be cut by any other plane passing 
through all its sides, the section will be an irregular 
hexagon. 

6th. The surface of the cube fig. 1, is shewn at fig. 
5, and if a piece of pasteboard be cut out, of that form, 
and cut half through in the lines crossing the figure, 
then folded together, it will form the regular solid. All 
the other solids may be made of pasteboard, in the 
same manner, if cut in the shape shewn in the cover- 
ings of the diagrams in the following plates. 

7th. The measure of the svrface of a cube is six 
times the square of one of its sides. Thus, if the side 
of a cube be one foot, the surface of one side will be 
one square foot, and its whole surface would be six 
square feet. 

Its solidity would be one cubic foot. 

Note. — ^The cube may also, m general, be called a prism, and 
a parallelopipedon, as it answers the description given of those 
bodies, but the terms are seldom applied to it. 



PLATE XV- 
SOLIDS AND THEIR COVERINGS, 

Fig. 1. Is a solid, bounded by six rectangular faces, 
each opposite pair being parallel, and equal to each 
other; the sides are oblong parallelograms, and the ends 



54 PLATE XV. 

are squares. It is called a right square prism, par- 

ALLELOPIPED, Or PARALLELOPIPEDON. 

Fig. 2. Is its covering stretched out. 

Fig. 3. Is a triangular prism ; its sides are rectan- 
gles, and its ends equal triangles. 

Fig. 4. Is its covering. 

Prisms derive their names from the shape of their 
ends, and the angles of their faces, thus: Fig. 1 is a 
square prism^ and fig. 2 a triangular prism. If the 
ends were pentagons, the prism would he pentagonal ; 
if the ends were hexagons, the prism would be hexa- 
gonal^ ^c. The sides of all regular prisms are equal 
rectangular parallelograms. 

Fig. 5. Is a square pyramid, bounded by a square 
at its base, and four regular triangles, as shewn at fig. 6. 

Pyramids, like prisms, derive their names from the 
shape of their bases ; thus we may have a square pyra- 
mid, as in fig. 4, or a triangular, pentagonal, or hexa- 
gonal pyramid, &c., as the base is a triangle, pentagon, 
hexagon, or any other figure. 

The sides of a pyramid incline together, forming a 
point at the top. This point is called its vertex^ apex^ 
or summit. 

The axis of a pyramid, is a line drawn from its sum- 
mit, to the centre of its base. The length of the axis, 
is the altitude of the pyramid. When the base of a 
pyramid is perpendicular to its axis, it is called a right 
pyramid; if they are not perpendicular to each other, 
the pyramid is oblique. If the top of a pyramid be cut 
off, the lower portion is said to be truncated ; it is also 
called dijrustrum of a pyramid, and the upper portion 
is still a pyramid, although only a segment of the ori- 
ginal pyramid. 

A pyramid may be divided into several truncated 
pyramids^ or frustrums^ and the upper portion remain a 



SOLIDS AND THEIR COVERINGS. 




Tiif. 3. 






If"'- Mimv. 



IJlnum k Sov^s 



Plate W 



SOLIBSAND TJIEDEVELOFEMENT OF THEIR SURFACES 



riq.l. 





liy.S 




Fuj.5 



Fuj.Z. 




ti/f. 4 




W"- Mai^'. 



Rljn/jn kSoii 



PLATE XV. 



55 



pyramid, as the name does not convey any idea of size, 
but a definite idea of form, viz: a solid, composed of 
an indefinite number of equal triangles, with their edges 
touching each other, forming a point at the top. 

A pyramid is said to be acute, right angled or ob- 
tuse, dependant on the form of its summit. 

An OBELISK is a pyramid whose height is very great 
compared to the breadth of the base. The top of an 
obelisk is generally truncated and cut off, so as to form 
a small pyramid, resting on the frustrum, which forms 
the lower part of the obelisk. 

When the polygon, forming the base of a pyramid, 
is irregular, the sides of the figure will be unequal, and 
the pyramid is called an irregulw pyramid. 



PLATE XVI. 

SOLIDS AND THEIR COVERINGS. 



Fig. 1. Is an hexagonal pyramid; and fig. 2 its 
covering. 

Fig. 3. A right cylinder, is bounded by tw^o uni- 
form circles, parallel to each other. The line connect- 
ing their centres, is called the axis. The sides of the 
cylinder is one uniform surface, connecting the circum- 
ferences of the circle, and everywhere equidistant from 
its axis. 



Problem 44. Fig. 4, 



To find the Length of the Parallelogram A. B. C. D, 
to form the Side of the Cylinder. 

1st. Draw the ends, and divide one of them into any 
number of equal parts, say twelve. 



56 



PLATE XVI. 



2nd. With the space of one of these parts, step off 
the same number on A, B, which will give the breadth 
of the covering to bend around the circles. 

Fig. 5. Is a right cone ; its base is a circle, its 
sides sloping equally from the base to its summit. A 
line drawn from its summit to the centre of the base, is 
called its axis. If the axis and base are not perpen- 
dicular to each other, it forms an oblique^ or scalene cone. 



Problem 45. Fig. 6. 



To draw the Covering. 

1st. With a radius equal to the sloping height of the 
cone, from E^ describe an indefinite arc, and draw the 
radius E. F. 

2nd. Draw^the circle of the base, and divide its cir- 
cumference into any number of equal parts, say twelve. 

3rd. With one of those parts in the dividers, step 
off from Fthe same number of times to G, then draw 
the radius E. G, to complete the figure. 



PLATE XVIL 
COVERINGS OF SOLIDS 



Fig. 1. The Sphere 



Is a solid figure presenting a circular appearance 
when viewed in any direction ; its surface is every 
where equidistant from a point within, called its centre. 

1st. It may be formed by the revolution of a semi- 
circle around its chord. 



ri/ite 17. 



SUBFACSS OF SOLIDS 



.%./. 



Figs 




Yii^' Mira/ie' . 



IUma.n EcSons. 



PLATE XVII. 57 

2nd. The chord around which it revolves is called 
the axis^ the ends of the axis are called poles, 

3rd. Any line passing through the centre of a sphere 
to opposite pointSj is called a diameter. 

4th. Every section of a sphere cut by a plane must 
be a circle, if the section pass through the centre, its 
section will be a great circle of the sphere ; any other 
section gives a lesser circle. 

5th. When a sphere is cut into two equal parts by a 
plane passing through its centre, each part is called a 
hemisphere ; any part of a sphere less than a hemisphere 
is called a segment ; this term may be applied to the 
larger portion as well as to the smaller. 

Problem 46. Fig. 2. 



To draw the Covering of the Sphere, 

1st. Divide the circumference into twelve equal parts. 

2nd. Step off on the line A, B the same number of 
equal parts, and with a radius of nine of those parts, 
describe arcs through the points in each direction ; 
these arcs will intersect each other in the lines C, D 
and E, F^ and form the covering of the sphere. 



Figure 3 



Is the surface of a regular Tetrahedron, it is 
bounded by four equal equilateral triangles. 

Figure 4. 



The regular Octahedron is bounded by eight equal 
equilateral triangles. 



58 PLATE XVII. 



Figure 5. 



The Dodecahedron is bounded by twelve equal 
pentagons. 

Figure 6. 



The IcosAHEDRON is bounded by twenty equal equi- 
lateral triangles. 

The four last figures, together with the hexahedron 
delineated on Plate 14, are all the regular polyhedrons. 
All the faces and all the solid angles of each figure are 
respectively equal. These solids are called platonic 
figures. 



PLATE XVIII. 

THE CYLINDER AND ITS SECTIONS. 



1st. If we suppose the right angled parallelogram 
A, B, C, -D, fig. 1, to revolve around the side A. J5, it 
would describe a solid figure ; the sides A, D and B. 
C would describe two circles whose diameters would 
be equal to twice the length of the revolving sides; the 
side C. D would describe a uniform surface connecting 
the opposite circles together throughout their whole 
circumference. The solid so described would be a 
right cylinder. 

2nd. The line A. JS, around which the parallelogram 
revolved, is called the axis of the cylinder, and as it 
connects the centres of the circles forming the ends of 
the cylinder, it is every where equidistant from its sides. 



Flate 16 



THE CYLINDER ANB SECTIONS . 



1) ! A 



<'L_ 




F^.l 



Fuj.Z. 



Fig. 3 . Sedioiv 

tliro . M. N. of Fig. 2 .^\ is 



FiifA.Sedujib 



H— 



'w 



Vr" Muafir 



rdirimi k Sons. 



PLATE XVIII. 



59 



3rd. If the ends of a cylinder be not at right angles 
to its axis, it is called an oblique cylinder. 

4th. If a cylinder be cut by any plane parallel to its 
axis, the section will be a parallelogram, as E. F. G. 
H, fig. 1. 

5th. If a cylinder be cut by any plane at right angles 
to its axis, the section will be a circle. 

6th. If a cylinder be cut by any plane not at right 
angles to its axis, passing through its opposite sides, 
as at K. L or M. JV, fig. 2, the section will be an el- 
lipsis, of which the line of section K. L or M. JV 
w^ould be the longest diameter, called the transverse 
or MAJOR diameter, and the diameter of the cylinder 
C. D would be the shortest diameter, called the con- 
jugate or minor diameter. 



Problem 47. Fig. 3. 



To describe an Ellipsis from the Cylinder with a string 

and two pins, 

1st. Draw the right lines JV. JM and C D at right 
angles to each other, cutting each other in S. 

2nd. Take in your dividers the distance P. M or P. 
JV, fig. 2, and set it off from S to M and JV, fig. 3, 
which will make M. JY equal to J\L JV, fig. 2. 

3rd. From ^, fig. 2, take J, D or ^. C, and set it 
off from S to Cand D, making C, I) equal to the di- 
ameter of the cylinder. 

4th. With a distance equal to 5'. M or S, JV from 
the points D and C, cut the transverse diameter in E 
and F; then E and F are the foci for drawing the 
ellipsis. 

Note. — E is a focus, and F is a focus. E and F are foct. 



60 PLATE XVIII. 

5th. In the foci, stick two pins, then pass a string 
around them, and tie the ends together at C 

6th. Place the point of a pencil at C, and keeping 
the string tight, pass it around and describe the curve. 

Note. — The surn of all lines drawn from the foci, to any point 
in the curve, is always constant and equal to the major axis : 
thus, the length of the lines E, R, and F, R, added together, is 
equal to the length of jG. C, and F. C, added together, or to two 
Hnes drawn from E and F, to any other point in the curve. 

7th. Fig. 4 is the section of the cylinder, through L. 
Ky fig. 2, and is described in the same way as fig. 3. 
The letters of reference are the same in both diagrams, 
except that the transverse diameter L. iC, is made 
equal to the line of section L. if, in fig. 2. 

8th. The line JV. M, fig. 3, or L, K, fig. 4, is called 
the TRANSVERSE, or MAJOR AXIS, (plural AXES,) and 
the line C, J), its conjugate, or minor axis. They 
are also called the transverse and conjugate diameters, 
as above defined. The transverse axis is the longest 
line that can be drawn in an ellipsis. 

9th. Any line passing through the centre S^ of an 
ellipsis, and meeting the curve at both extremities, is 
called a diameter: every diameter divides the ellipsis 
into two equal parts. The conjugate of any diame- 
ter, is a line drawn through the centre, terminated by 
the curve, parallel to a tangent of the curve at the ver- 
tex of the said diameter. The point where the diame- 
ter meets the curve, is the vertex of that diameter. 

10th. An ordinate to any diameter, is aline drawn 
parallel to its conjugate, and terminated by the curve 
and the said diameter. An abscissa is that portion 
of a diameter intercepted between its vertex and ordi- 
nate. Unless otherwise expressed, ordinates are in 
general, referred to the axis, and taken as perpendicular 
to it. Thus, in fig. 4, X Fis the ordinate to, and L. 



PLATE XIX. 61 

Xand K, X, the abscissae of the axis K, L. — F. W is 
the ordinate to, and C F, and D. F, the abscissae of 
the axis C D. 



PLATE XIX. 

THE CONE AND ITS SECTIONS. 

Definitions. 



1st. A CONE is a solid, generated by the revolution 
of a right angled triangle about one of its sides. 

2nd. If both legs of the triangle are equal, as S, JY 
and JY, 0, fig. 2, it would generate a right angled 
CONE ; the angle S being a right angle. 

3rd. If the stationary side of the triangle be longest, 
as M. JYj the cone will be acute, and if shortest, as 
T. JSr^ it will be obtuse angled. 

4th. The base of a cone is a circle, from which the 
sides slope regularly to a point, which is called its 
vertex, apex, or summit. 

5th. The axis of a cone, is a line passing from the 
vertex to the centre of the base, as M. JV, figs. 1, 2, 
3 and 4, and represents the line about which the tri- 
angle is supposed to rotate. 

6th. A right cone. When the axis of a cone is 
perpendicular to its base, it is called a right cone ; if 
they are not perpendicular to each other, it is called an 
oblique cone. 

7th. If a cone be cut by a plane passing through its 
vertex to the centre of its base, the section will be a 
triangle. 



62 PLATE XIX. 

8th. If cut by a plane, parallel to its base, the sec- 
tion will be a circle, as at E, F, fig. 1. 

9th. If the upper part of fig. 1 should be taken 
away, as at E, JP, the lower part would be a trunca- 
ted CONE, or FRUSTRUM, the part above E. F^ w^ould 
still be a cone; and, if another portion of the top were 
cut off from it, another truncated cone would be form- 
ed : thus a cone may be divided into several truncated 
cones, and the portion taken from the summit, would 
still remain a cone. Similar remarks have already 
been applied to the pyramid. 

10th. If a cone be cut by any plane passing through 
its opposite sides, as at ^. JB, fig. 3, the section will 
be ^n ellipsis, 

11th. If a cone be cut by a plane, parallel with one 
of its sides, as at P. Q., R, 5', or RJ SJ fig. 4, the 
section will be a parabola. 

12th. If a cone be cut by a plane, which, if con- 
tinued, would meet the opposite cone, as through C 
D, fig. 4, meeting the opposite cone at 0, the section 
will be an hyperbola. 



Problem 48. Fig. 3. 



To describe the Ellipsis from the Cone. 

1st. Let fig. 3 represent the elevation of a right cone, 
and Jl. B the line of section. 

2nd. Bisect A. B in C 

3rd. Through C, draw E. F perpendicular to the 
axis Jkf. JV*, cutting the axis in P. 

4th. With one foot of the dividers in P, and a radius 
equal to P. E, or P. P, describe the arc E. D, F. 

5th. From C, the centre of the line of section j1. P, 



THE CO YE AND ITS SECTIONS. 



M 




Fi^.3. 




^i 



:B: 








^^i. F^y- 




IT'-Mmwr 



iZ/'/.'w// ,?- .C« 



PLATE XIX. 



63 



draw C D parallel to the axis, cutting the arc E. D. F, 
in D. 

6th. Then ji. B is the transverse axis^ and C D its 
semiconjugate of an, ellipsis, which may be described 
with a string, as explained for the section of the cylin- 
der, or by any of the other methods to be hereafter de- 
scribed. 

Note. — A section of the cylinder, as well as of a cone, passing 
through its opposite sides, is always an ellipsis. In the cone, 
the length of both axes vary with every section, but in the cylin- 
der, the conjugate axis is always equal to the diameter of the 
cylinder, whatever may be the inclination of the line of section. 



Problem 49. Fig. 4. 



To find the length of the base line for describing the 

other sections. 

1st. With one foot of the dividers m JV, and a ra- 
dius equal to JV. T, or JV. F, describe a semicircle, 
equal to half the base of the cone. 

2nd. From C and P, the points where the sections 
intersect the base, draw P. A, and C JB, cutting the 
semicircle in A and J5. Then A. P is one-half the 
base of the parabola, and C B is one-half the base of 
the hyperbola. The methods for describing these curves, 
are shewn in Plates 20 and 21. 



64 PLATE XX. 



PLATE XX. 
TO DESCRIBE THE ELLIPSIS AND HYPERBOLA. 

Problem 50. Fig. 1. 



To find Points in the Curve of an Ellipsis by In- 
tersecting Lines. 

Let Jl.By be the given transverse axis, and C. D, the 
conjugate. 

1st. Describe the parallelogram L. M, JV. 0, the 
boundaries passing through the ends of the axes. 

2nd. Divide A, L,—A. JY,—B. Jlf,— and 5. 0, into 
any number of equal parts, say 4, and number them as 
in the diagram. 

3rd. Divide A. S, and B. S, also into 4 equal parts, 
and number them from the ends toward the centre. 

4th. From the divisions in A. L and jB. JIf, draw 
lines to one end of the conjugate axis at C; and, from 
the divisions in jB. and A. JV, draw lines to the other 
end at D. 

5th. From D, through the points 1, 2, 3, in A. S, 
draw^ lines to intersect the lines 1, 2, 3 drawn from the 
divisions on A, i, and in like manner through B. S, 
to intersect the lines from B. M, These points of in- 
tersection are points in the curve. 

6th. From C, through the divisions 1, 2, 3, on S, A 
and S, J5, draw lines to intersect the lines 1, 2, 3 drawn 
from A. JV and jB. O, which will give the points for 
drawing the other half of the curve. 



m 



Plat3 20. 



ELLIPSIS .dJm HYPERBOLA 



Fi/].l 




W'-MimOe.. 



IHtnan Ec S nns 



PLATE XX. 



65 



7lh. Through the points of intersection, trace the 
curve. 

Note. — If required on a small scale, the curve can be drawn 
by hand ; but, if required on a large scale, for practical purposes, 
it is best to drive sprigs at the points of intersection, and bend a 
thin flexible strip of pine around them, for the purpose of tracing 
the curve. Any number of points may be found by dividing the 
hues into the requisite number of parts. 



Figure 4. 



Is a semi-ellipsis, drawn on the conjugate axis by 
the same method, in which Jl, B is the transverse, and 
C. D the conjugate axis. 

Note. — ^This method will apply to an ellipsis of any length or 
breadth. 



Problem 51. Fig. 2. 



To draw an Ellipsis with a Trammel, 

The TRAMMEL shcwn in the diagram is composed of 
two pieces of wood halfened together at right angles to 
each other, with a groove running through the centre 
of each, the groove being wider at the bottom than at 
the top. J. K, L is another strip of wood with a point 
at J, or with a hole for inserting a pencil at J, and two 
sliding buttons at jS^and L ; the buttons are generally 
attached to small morticed blocks sliding over the strip, 
with wedges or screw^s for securing them in the proper 
place ; (the pins are only shewn in the diagram,) the 
buttons attached to the pins are made to slide freely in 
the grooves. 



66 PLATE XX. 

Mode of Setting the Trammel. 

1st. Make the distance L K equal to the semi-con- 
jugate axis, and the distance from I to L equal to the 
semi-transverse axis. 

2nd. Set the grooved strips to coincide with the 
axes of the ellipsis, and secure them there. 

3rd. Move the point I around and it will trace the 
curve correctly. 

Note. — This is a very useful instrument, and was formerly 
used very frequently by carpenters to lay off their work, and also 
by plasterers to run their mouldings around elliptical arches, &c., 
the mould occupying the position of the point /. It was rare 
then to find a carpenter's shop without a trammel or to find a 
good workman who was not skilled in the use of it ^ but since 
Grecian architecture with its horizontal lintels has taken the place 
of the arch, it is seldom a trammel is required, and when required, 
much more rare to find one to use; but as it is sometimes wanted, 
and few of our young mechanics know how to apply it, at the 
risk of being thought tedious, we have been thus minute in its 
description. 



Problem 52. Fig. 3. 



To describe the Hyperbola from the Cone, 

1st. Draw the line A. C. B and make C. JS and C. 
A each equal to C, J5, fig. 4, plate xix, then A. B will 
be equal to the base of the hyperbola. 

2nd. Perpendicular to A. B^ draw A. E and B. F^ 
and make them equal to C. D, fig. 4, plate xix. 

3rd. Join E. F, from C erect a perpendicular C. D. 
0, and make C. equal to C. 0, fig. 4. plate xix. 

4th. Divide A. E and jB. F each into any number 
of equal parts, say 4, and divide B. C and C. A into 
the same number, and number them as in the diagram. 



PLATE XXi 



67 



5th. From the points of division on A, E and B. 
jP, draw lines to D. 

6th. From the points of division in Jl. B, draw lines 
toward 0, and the points where they intersect the other 
lines with the same numbers will be points in the curve. 
The curve A. D. B is the section of the cone through 
the line C. D, fig. 4, plate xix. 



PLATE XXI. 
PARABOLA AND ITS APPLICATION, 



Problem 53. Fig. 1 



To describe the Parabola by Tangents. 

1st. Draw A, P. B, make A, P and P. B each equal 
to A, Py fig. 4, plate xix. 

2nd. From P draw P. Q. R perpendicular to A, B^ 
and make P. R equal to twice the height of P. Q, fig. 
4, plate xix. 

3rd. Draw A. R and B. P, and divide them each 
into the same number of equal parts, say eight ; num- 
ber one side from A to P, and the other side from R 
to P. 

4th. Join the points 1. 1. — 2. 2. — 3. 3, &c.; the 
lines so drawn will be tangents to the curve, which 
should be traced to touch midway between the points 
of intersection. 

The curve A, Q. P is a section of the cone through 
P. Q, fig. 4, plate xix. 



68 PLATE XXI. 



Problem 54. Fig. 2. 



To describe the Parabola by another method. 

Let A, B be the width of the base and P. Q the 
height of the curve. 

1st. Construct the parallelogram A. B. C, D. 

2nd. Divide A. C and A. P—P. B and B. D re- 
spectively into a similar number of equal parts ; number 
them as in the diagram. 

3rd. From the points of division in A. C and JB. D, 
draw lines to Q. 

4th. From the points of division on A. B erect per- 
pendiculars to intersect the other lines ; the points of 
intersection are points in the curve. 



Problem 55. Fig. 3. 



To describe a Parabola by continued motion^ with a 
RuleVy String and Square. 

Let C. D be the width of the curve and H. J the 
height. 

1st. Bisect H. D in if, draw J. jfiCand K. E perpen- 
dicular to /. K, cutting J. H extended in E. Then 
take the distance H. E and set it off from / to F, then 
F is the focus. 

2nd. At any convenient distance above J, fasten a 
ruler A. B^ parallel, to the base of the parabola C D. 

3rd. Place a square S^ with one side against the 
edge of the ruler, A. B, the edge 0. JV of the square 
to coincide with the line E. J. 

4th. Tie one end of a string to a pin stuck in the 
focus at jP, place your pencil at J, pass the string 



Flat^ 21 . 



P^RjlBOLJi 



ri/].i 




Fuj.Z. 



^r I ^' ^-^. 



3 4-3?.! 



w=^m 


^-^—=- = — - ^-^^-^'=--^---_-.- _- __-_- -_^ .- , 




s 


'.] 


r 


Fuj.S. 






\ 




H 


5 





1 1 3 F 3 2 1 



//^ 


3 


. \ 




\ 


/ /y/ 






1 / / 


2 






M 


\/f / 






\ ^ 


n / 


1 




\ 


,\\\ 


/// 








' \\ 


r 








1 



Vl'^-Mui^^ 



ninuuiSj. 



PLATE XXI. 



69 



around it, and bring it down to JV, the end of the 
square, and fasten it there. 

5th. With the pencil at J, against the side of the 
square, and the string kept tight, slide the square along 
the edge of the ruler towards A ; the pencil being kept 
against the edge of the square, with the string stretched, 
will describe one half of the parabola, J, C. 

6th. Turn the square over, and draw the other half 
in the same manner. 



Definitions. 



1st. The FOCUS of a parabola is the point jP, about 
which the string revolves. The edge of the ruler ^. 
By is the directrix of the parabola. 

2nd. The axis is the line /. i/, passing through the 
focus, and perpendicular to the base C. D. 

3rd. The principal vertex, is the point /, where 
the top of the axis meets the curve. 

4th. The parameter, is a line passing through the 
focus, parallel to the base, terminated at each end by 
the curve. 

5th. Any line, parallel to the axis, and terminating 
in the curve, is called a diameter, and the point where 
it meets the curve, is called the vertex of that diameter. 



Problem 56. Fig. 4, 



To apply the Parabola to the construction of Gothic 

Arches, 

1st. Draw A. JB, and make it equal to the width of 
the arch at the base. 

2nd. Bisect A. B in E^ draw E. F perpendicular to 
A. By and make E. jP equal to the height of the arch. 



70 PLATE XXI. 

3rd. Construct the parallelogram A. B. C. D. 

4th. Divide E. F into any number of equal parts, 
and D. F and jP. C each into a similar number, and 
number them as in the diagram. 

5th. From the divisions on jP. D, draw lines to A, 
and from the divisions on F. C, draw lines to B. 

6th. Through the divisions on E. jP, draw lines par- 
allel to the base, to intersect the other lines drawn from 
the same numbers on D. C. The points of intersection 
are points in the curve through which it may be traced. 

Note. — If we suppose this diagram to be cut through the line 
E. F, and turned around until E. Jl and E, B coincide, it will 
form a parabola, drawn by the same method as fig. 2; and, if we 
were to cut fig. 2 by the line P. Q, and turn it around until P. 
JL and P. B coincide, it would form a gothic arch, described by 
the same method as fig. 4 ; and, if the proportions of the two 
figures were the same, the curves would exactly coincide. 



PLATE XXII. 

Problem 57. Fig. 1 



To draw the Boards for covering Circular Domes. 

To lay the boards vertically. Let A, D. C be half 
the plan of the dome ; let D. C represent one of the 
ribs, and E. jPthe width of one of the boards. 

1st. Draw D. 0, and continue the line indefinitely 
toward H. 

2nd. Divide the rib D. C into any number of equal 
parts, and from the points of division, draw lines 
parallel to A. C, meeting B. in 1, 2, 3, &c. 

3rd. With an opening of the dividers equal to one 
of the divisions on D. C, step off from D toward jH, 



TO DRAW THE BOARBS FOlt COVERING 
HEMISPHERICAL DOMES. 



Fig.l. 




f— r d ^ 




Fl^.2 




W"- MiiLLfi.. 



RhTuai %cjo-n.s. 



PLATE XXII. 



71 



the same number of parts as D. C is divided into, 
making the right line D. Hj nearly equal to the curve 

D. a 

4th. Join E. and F. 0. 

5th. Make 1. c — 2. d — 3. e — 4. / and 5 g, on each 
side of D. H, equal to 1. c — 2. d — 3. e, &c. on D. 0. 

6th. Through the points c. d. e,f.gy trace the curve, 
which v^ill be an arc of a circle ; and if a series of 
boards made in the same manner, be laid on the dome, 
the edges will coincide. 

Note. — In practice where much accuracy is required, the rib 
should be divided into at least twelve parts. 



Problem 58. Fig. 2, 



To lay the boards Horizontally. 

Let ^. jB. C be the vertical section of a dome 
through its axis. 

1st. Bisect Ji. C in D, and draw D. P perpendicu- 
lar to ji. a 

2nd. Divide the arc ^. B into such a number of 
equal parts, that each division may be less than the 
breadth of a board. (If we suppose the boards to be 
used, to be of a given length, each division should be 
made so that the curves struck on the hollow side 
should touch the ends, and the curves on the convex 
side should touch the centre.) 

3rd. From the points of division, draw lines parallel 
to JI. C to meet the opposite side of the section. Then 
if we suppose the curves intercepted by these lines to 
be straight lines, (and the difference will be small,) 
each space w^ould be the frustrum of a cone, whose 
vertex would be in the line D. P, and the vertex of 



72 PLATE XXII. 

each frustrum would be the centre from which to de- 
scribe the curvature of the edges of the board to fit it. 

4th. From 1 draw a line through the point 2, to 
meet the line D. P in E ; then from £, with a radius 
equal io E. 1, describe the curve 1. L, which will give 
the lower edge of the board, and with a radius equal 
to E, 2, describe the arc 2. JK", which will give the 
upper edge. The line L. K drawn to J5, will give the 
cut for the end of a board which will fit the end of any 
other board cut to the same angle. 

5th. From 2 draw a line through 3, meeting D. P in 

F. From 3, draw a line through 4, meeting D. P in 

G, and proceed for each board, as in paragraph 4. 
6th. If from C we draw a line through Jkf, and con- 
tinue it upward, it would require to be drawn a very 
great distance before it would meet D, P ; the centre 
would consequently be inconveniently distant. 

For the bottom board, proceed as follows : 

1st. Join A, M, cutting D. P in JV, and join JV. 1. 

2nd, Describe a curve, by the methods in Problems 

34 or 35, Plate 11, through 1. JV, Jkf, which will give 

the centre of the board, from which the width on either 

side may be traced. 



PLATE XXIII. 
CONSTRUCTION OF ARCHES 

Arches in architecture are composed of a number of 
stones arranged symmetrically ov^ an opening intend- 
ed for a door, window, &c., for the purpose of support- 
ing a superincumbent weight. The depth of the stones 
are made to vary to suit each particular case, being 



PLATE XXIII. 



73 



made deeper in proportion as the width of an opening 
becomes larger, or as the load to be supported is in- 
creased ; the size of the stones also depends much on 
the quality of the material of which they are composed : 
if formed of soft sandstone they will require to be much 
deeper than if formed of granite or some other hard 
strong stone. 



Definitions. Fig. 2. 



1st. The SPAN of an arch is the distance between the 
points of support, which is generally the width of the 
opening to be covered, as A. B. These points are 
called the springing points; the mass against which the 
arch rests is called the abutment. 

2nd. The rise, height or versed sine of an arch, 
is the distance from C to D. 

3rd, The springing line of an arch is the line A, 
B^ being a horizontal line drawn across the tops of the 
support where the arch commences. 

4th, The crown of an arch is the highest point, as D. 

5th. VoussoiRS is the name given to the stones 
forming the arch. 

6th. The keystone is the centre or uppermost 
voussoir D, so called, because it is the last stone set, 
and wedges or keys the whole together. Keystones 
are frequently allowed to project from the face of the 
wall, and in some buildings are very elaborately sculp- 
tured. 

7th. The intrados or soffit of an arch is the 
under side of the voussoirs forming the curve. 

8th. The extrados or back is the upper side of the 
voussoirs. 

9th. The thrust of an arch is the tendency which 



74 PLATE XXIII. 

all arches have to descend in the middle, and to over- 
turn or thrust asunder the points of support. 

Note. — ^The amount of the thrust of an arch depends on the 
proportions between the rise and the span, that is to say, the span 
and weight to be supported being definite; the thrust will be di- 
minished in proportion as the rise of the arch is increased, and 
the thrust will be increased in proportion as the crown of the 
arch is lowered. 

10th. The JOINTS of an arch are the lines formed by 
the adjoining faces of the voussoirs; these should gen- 
erally radiate to some definite point, and each should 
be perpendicular to a tangent to the curve at each joint. 
In all curves composed of arcs of circles, a tangent to 
the curve at any point would be perpendicular to a ra- 
dius drawn from the centre of the circle through that 
point, consequently the joints in all such arches should 
radiate to the centre of the circle of which the curve 
forms a part. 

11th. The BED of an arch is the top of the abut- 
ment; the shape of the bed depends on the quality of 
the curve, and will be explained in the diagrams. 

12th. A RAMPANT ARCH is ouc in which the spring- 
ing points are not in the samp level. 

13th. A STRAIGHT ARCH, or as it is more properly 
called, a plat band, is formed of a row of wedge- 
shaped stones of equal depth placed in a horizontal line, 
the upper ends of the stones being broader than the 
lower, prevents them from falling into the void below. 

14th. Arches are named from the shape of the curve 
of the under side, and are either simple or complex. I 
would define simple curves to be those that are struck 
from one centre, as any segment of a circle, or by con- 
tinued motion, as the ellipsis^ parabola, hyperMa, cy- 
cloid and epicycloid; and complex arches to be those 
described from two or more fixed centres, as many of 



PLATE XXIII. 



75 



the Gothic or pointed arches. The simple curves have 
all been described in our problems of practical Geome- 
try ; we shall however repeat some of them for the pur- 
pose of shewing the method of drawing the joints. 



Problem 59. Fig. 1 



To describe a Segment o?* Scheme Arch^ and to draw the 

Joints, 

1st. Let E and F be the abutments, and the 
centre for describing the curve. 

2nd. With one foot of the dividers in 0, and the 
distance 0. jF, describe the line of the intrados. 

3rd. Set off the depth of the voussoirs, and with the 
dividers at 0, describe the line of the extrados. 

4th. From JG and jF draw lines radiating. to 0, which 
gives the line of the beds of the arch. This line is often 
called by masons a skew-hack. 

5th. Divide the intrados or extrados, into as many 
parts as you design to have stones in the arch, and ra- 
diate all the lines to 0, which will give the proper di- 
rection of the joints, 

6th. If the point should be at too great distance 
to strike the curve conveniently, it may be struck by 
Problem 34 or 35, Plate 11; and the joints may be 
found as follows : Let it be desired to draw a joint at 
2, on the line of the extrados ; from 2 set off any dis- 
tance on either side, as at 1 and 3 ; and from 1 and 3, 
with any radius, draw two arcs intersecting each other 
at 4 — then from 4 through 2 draw the joint which will 
be perpendicular to a tangent, touching the curve at 2, 
This process must be repeated for each joint. The 
keystone projects a little above and below the lines of 
the arch. 



76 PLATE XXIII, 



Prob. 60. Fig. 2.— The Semicircular Arch, 



This requires but little explanation. Jl. B is the 
span and C the centre, from which the curves are struck, 
and to which the lines of all the joints radiate. The 
centre C being in the springing line of the arch the beds 
are horizontal. 



Prob. 61. Fig. 3. — The Horse-shoe Arch, 



Is an arc of a circle greater tlian a semicircle, the 
centre being above the springing line. 

This arch is also called the Saracenic orMoRESco 
arch, because of its frequent use in these styles of arch- 
itecture. The joints radiate to the centre, as in fig. 2. 
The joint at 5, below the horizontal line, also radiates 
to 0. This may do very well for a mere ornamental 
arch, that has no weight to sustain; but if, as in the 
diagram, the first stone rests on a horizontal bed, it 
would be larger on the inside than on the outside, and 
would be liable to be forced out of its position by a 
slight pressure, much more so than if the joint were 
made horizontal, as at 6. These remarks will also 
apply to fig. 4, Plate 24. 



Problem 62. Fig. 4. 



To describe an Ogee Arch^ or an Arch of Contrary 

Flexure. 

Note. — ^This arch is seldom used over a large opening, but 
occurs frequently in canopies and tracery in Gothic architecture, 
the rib of the arch being moulded. 

1st. Let A, B be the outside width of the arch, and 

C. B the height, and let A. E be the breadth of the rib. 



FlaU 23 . 
JOINTS IN JinCHES 



fuf.l. 



ry.z. 





W" Mniifii 



I in an ^r ,s,->ns 



PLATE XXIir, 



77 



2nd. Bisect A. B in C, and erect the perpendicular 
C. D; bisect A. C in JP, and draw J^. / parallel to 
a D. 

3rd. Through D draw J. K parallel to A. By and 
make D. K equal to D. J. 

4th. From F set off F. G, equal to ^. E the breadth 
of the rib, and make C Jf equal to C G. 

5th. Join G. J and jEf. K; then G and iZ" will be the 
centres for drawing the lower portion of the arch, 
J and K will be the centres for describing the upper 
portion, and the contrary curves will meet in the lines 
G.J^ndKK. 



Problem 63. Fig. 5. 



To draw the Joints in an Elliptic Arch. 

Let A. B be the span of the arch, C D the rise, and 
jF. F the foci, from which the line of the intrados may 
be described. 

The voussoirs near the spring of the arch are in- 
creased in depth, as they have to bear more strain than 
those nearer the crown ; the outer curve is also an 
ellipsis, of which H and H are the foci. 

To draw a joint in any part of the curve, say at 5. 

1st. From F and F the foci, draw^ lines cutting each 
other in the given point 5, and continue them out in- 
definitely. 

2nd. Bisect the angle 5 by Problem 11, Page 26, 
the line of bisection will be the line of the joint. 

The joints are found at the points 1 and 3 in the 
same manner. 

3d. If we bisect the internal angle, as for the joints 
2 and 4, the result will be the same. 



78 



PLATE XXIII, 



4th. To draw the corresponding joints on the oppo- 
site side of the arch, proceed as follows : — 

5th. Prolong the line C, D indefinitely toward JC, 
and prolong the lines of bisection 1, 2, 3, 4 and 5, to 
intersect C jG in 1, 2, 3, &c.j and from those points 
draw the corresponding joints between A and D. 



PLATE XXIV. 



TO DESCRIBE GOTHIC ARCHES AND TO DRAW 

THE JOINTS. 



The most simple form of Pointed or Gothic arches 
are those composed of two arcs of circles, whose cen- 
tres are in the springing line. 

Figure 1. — The Lancet Arch. 



When the length of the span A. B is much less than 
the length of the chord A, C, as in the diagram, the 
centres for striking the curves will be some distance 
beyond the base, as shewn by the rods ; the joints all 
radiate to the opposite centres. 



Fig. 2. — The Equilateral Arch. 



When the span D. jE, and the chords D. F and E. 
F form an equilateral triangle, the arch is said to be 
equilateral, and the centres are the points B and E in 
the base of the arch, to which all the joints radiate. 



Fhcte^ Z4 



P OUST TED Jilt CHE S . 



Fi^.J. 



Fuj.^ 




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Ulm.av. 3c Sons. 



PLATE XXIV. 



79 



Fig. 3. — The Depressed Arch 



Has its centres within the base of the arch, the chords 
being shorter than the span ; the joints radiate to the 
centres respectively. 

Note. — ^There are no definite proportions for Gothic arches, 
except for the equilateral ; they vary from the most acute to those 
whose centres nearly touch, and which deviate but little from a 
semicircle. 



Fig. 4. — The Pointed Horse-shoe Arch. 



This diagram requires no explanation; the centres 
are above the springing line. See fig. 3, plate xxiii, 
page 76. 



Figure 5, 



To describe the Four Centred Pointed Arch* 

1st. Let Jl, B be the springing line, and E. C the 
height of the arch. 

2nd. Draw J5. D parallel to E. C\ and make it equal 
to two-thirds of the height of E, C\ 

3rd. Join D. C, and from C draw C. L perpendicu- 
lar to a D. 

4th. Make C. G and B. F both equal to J5. D. 

5th. Join G. F, and bisect it in H, then through H 
draw H, L perpendicular to G. F meeting C. L in L, 

6th. Join L, F, and continue the line to JY. Then 
L and F are the centres for describing one-half of the 
arch, and the curves will meet in the line L. F, JY. 

7th. Draw L. M parallel to Jl. JS, make 0. M equal 
to 0. L, and E. K equal to E. F. Then /iTand Jkf are 



80 



PLATE XXIV. 



the centres for completing the arch, and the curves will 
meet in the line M. K. P. 

8th. The joints from P to C will radiate to M ; from 
C to JV they will radiate to L ; from JV to jB they will 
radiate to F^ and from P to A they will radiate to K, 

Note 1. — ^As the joint at P radiates to both the centres iTand 
M, and the joint at JV radiates both to F and L, the change of 
direction of the lower joints is easy and pleasing to the eye, so 
much so that we should be unconscious of the change, if the 
constructive lines were removed. 

Note 2. — When the centres for striking the two centered arch 
are in the springing line, as in diagrams 1 , 2 and 3, the vertical 
side of the opening joins the curve, without forming an unpleas- 
ant angle, as it would do if the vertical Hnes were continued up 
above the line of the centres; it is true that examples of this char- 
acter may be cited in Gothic buildings, but its ungraceful appear- 
ance should lead us to avoid it. 



PLATE XXV. 
OCTAGONAL PLAN AND ELEVATION. 

Fig. 1. — Half the Plan. Fig. 2. — Elevation, 



This plate requires but little explanation, as the 
dotted lines from the different points of the plan, per- 
fectly elucidate the mode of drawing the elevation. 

The dotted line A^ shews the direction of the rays 
of light by which the shadows are projected; the mode 
of their projection will be explained in Plates 47 and 
48. 



FhiM 25 



OCTAGONAL PLAN AND ELEVATION 




rilmar.l'' So, 



7V/y//' ^/; 



riB(-ULAR FL^4JSr yiND ELET'jlTION 




PLATE XXVU 



81 



PLATE XXVI. 
CIRCULAR PLAN AND ELEVATION. 



This plate shews the mode of putting circular objects 
in elevation. The dotted lines from the different points 
of the plan, determine the widths of the^amJ^ (sides) 
of the door and windows, and the projections of the 
sills and cornice. One window is farther from the 
door than the other, for the purpose of shewing the 
different apparent widths of openings, as they are more 
or less inclined from the front of the picture. 

This, as well as Plate 25, should be drawn to a 
much larger size by the learner; he should also vary 
the position and width of the openings. As these de- 
signs are not intended for a particular purpose, any 
scale of equal parts may be used in drawing them. 



PLATE XXVII. 
ROMAN MOULDINGS. 



Roman mouldings are composed of straight lines and 
arcs of circles. 

Note. — Each separate part of a moulding, and each moulding 
in an assemblage of mouldings, is called a member. 



Fig. 1. — A Fillet, Band or Listel 



Is a raised square member, with its face parallel to 
the surface on which it is placed. 



82 



PLATE XXVII, 



Fig. 2. — Bead. 



A moulding whose surface is a semicircle struck from 
the centre K. 



Fig. 3. — Torus. 



Composed of a semicircle and a fillet. The projec- 
tion of the circle beyond the fillet, is equal to the ra- 
dius of the circle, which is shewn by the dotted line 
passmg through the centre L, The curved dotted line 
above the fillet, and the square dotted line below the 
circle, shew the position of those members when used 
as the base of a Doric column. 



Figure 4. — The Scotia 



Is composed of two quadrants of circles between 
two fillets. B is the centre for describing the large 
quadrant ; A the centre for describing the small quad- 
rant. The upper portion may be made larger or smaller 
than in the diagram, but the centre A must always be 
in the line B. A. The scotia is rarely, if ever used 
alone ; but it forms an important member in the bases 
of columns. 



Fig. 5. — The Ovolo 



Is composed of a quadrant between two fillets. C 
is the centre for describing the quadrant. The upper 
fillet projects beyond the curve, and by its broad 
shadow adds much to the effect of the moulding. The 



Flate Z7. 



ROMJLjy MOULD lA^GS 



FVlet 



Fu^.l. 



Fiq.:^. 



L 



B>uj^7 




Fuf . 3 



Toni^ 



Fi^. 4. Scolia 




Fii) 


. 3 . 


//jv^7/^ 


f 




\ 


H 


<^rt 



Fi^ . 6. Q:a'eM7? 




Fig . 7. Cyma Fecta. Fig . 6 . Cyma ReverscL 




W"l^i>>:fi, 



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PLATE XXVII. 



83 



ovolo is generally used as a bed moulding, or in some 
other position where it supports another member. 



Fig. 6. — The Cavetto, 



Like the ovolo, is composed of a quadrant and two 
fillets. The concave quadrant is used for the cavetto 
described from D ; it is consequently the reverse of the 
ovolo. The cavetto is frequently used in connection 
with the ovolo, from which it is separated by a fillet. 
It is also used sometimes as a crown moulding of a 
cornice ; the crown moulding is the uppermost member. 



Fig. 7. — The Cyma Recta 



Is composed of two arcs of circles forming a waved 
line, and two fillets. 

To describe the cyma, let I be the upper fillet and 
J\r the lower fillet. 

1st. Bisect /. JV. in M, 

2nd. With the radius M. JV or M, I, and the foot 
of the dividers in JVand Jkf, successively describe two 
arcs cutting each other in F, and from M and I with 
the same radius, describe two arcs, cutting each other 
in E. 

3rd. With the same radius from E and JP, describe 
two arcs meeting each other in M. 

The proportions of this moulding may be varied at 
pleasure, by varying the projection of the upper fillet. 



Fig. 8. — The Cyma Reversa, Talon or Ogee 



Like the cyma recta, it is composed of two circular 



84 PLATE XXVII. 

arcs and two fillets ; the upper fillet projects beyond 
the curve, and the lower fillet recedes within it. 

The curves are described from G and H. 

The CYMA, or cyma recta has the concave curve 
uppermost. 

The CYMA REVERSA has the concave curve below. 

The CYMA RECTA is uscd as the upper member of an 
assemblage of mouldings, for which it is well fitted 
from its light appearance. 

The CYMA REVERSA from its strong form, is like the 
ovolo, used to sustain other members. 

The dotted lines drawn at an angle of 45^ to each 
moulding, shew the direction of the rays of light, from 
which the shadows are projected. 

Note. — ^When the surface of a moulding is carved or sculptur- 
ed, it is said to be enriched. 



PLATE XXVIIL 



GRECIAN MOULDINGS 

Are composed of some of the curves formed by the 
sections of a cone, and are said to be elliptic, parabolic, 
or hyperbolic, taking their names from the curves of 
which they are formed. 



Figures 1 and 2. 



To draw the Grecian Echinus or Ovolo, the fillets A 
and B, the tangent C. B, and the point of greatest 
projection D being given, 

1st. Draw B, H^ a continuation of the upper edge 
of the under fillet. 



Flatp 2.8 



GRECIAN MOULDINGS 



Echinus or Ovolo 




Fuf.l 




Fiq . Z 



Cym/i 



Be/^a, 




E D 



G <H 



C F B 



Fig .3 



K 




Cvni/x Fever.'i 



Sr/ytuf 




Fifj . 5 . 



1 2 S 4^ 




Fin.n 



12 3^ 



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Illwan k Sons 



PLATE XXVIII. 



85 



2nd. Through D, draw D. H perpendicular to B. Hy 
cutting the tangent B. C in C 

3rd. Through J5, draw B. G parallel to D. H, and 
through D, draw JD. E parallel to H. By cutting G. B 
inE. 

4th. Make E. G equal to E. By and E. F equal to 
H. Cy join jD. F. 

5th. Divide the lines D. F and D* C each into the 
same number of equal parts. 

6th. From the point By draw lines to the divisions 
1,2, 3, &c. in D. C. 

7th. From the point Gr, draw lines through the divi- 
sions in jD. Fy to intersect the lines drawn from B, 

8th. Through the points of intersection trace the 
curve. 

Note. — A great variety of form may be given to the echinus, 
by varying the projections and the inclination of the tangent jB. C. 

Note 2. — If H, C is less than C. D, as in fig. 1, the curve 
will be elliptic ; if H. C and C. D are equal, as in fig. 2, the 
curve is parabolic; if H. C be made greater than D, C, the curve 
will be hyperbolic. 

Note 3. — ^The echinus, when enriched with carving, is gener- 
ally cut into figures resembling eggs, with a dart or tongue be- 
tween them. 



Figs. 3 and 4* — The Grecian Cyma 



To describe the Cyma Rectay the perpendicular height 
B. D and the projection A. D being given, 

1st. Draw A. C and B. D perpendicular to A, D 
and a B parallel to A. B 

2nd. Bisect Jl. D in E, and A, C in G; draw E. 
F and G. 0, which will divide the rectangle A. C. B 
D into four equal rectangles. 



86 



PLATE XXVIII. 



3rd. Make G. P and 0. K each equal to 0. H. 

4th. Divide A. G—0. B—A. E and J5. jF into a 
similar number of equal parts. 

5th. From the divisions in A. E and F. jB, draw 
lines to H; from P draw lines through the divisions 
on A. G to intersect the lines drawn from A. J5, and 
from K through the divisions in O. JB, draw lines to 
intersect the lines drawn from jP. B. 

6th. Through the points of intersection draw the 
curve. 

Note. — ^I'he curve is formed of two equal converse arcs of an 
ellipsis, of which E, F is the transverse axis, and P. Hoi H. K 
the conjugate. The points in the curve are found in the same 
manner as in fig. 1, plate 20. 



Fig. 5. — The Grecian Cyma Reversa, Talon or 

Ogee. 



To dram the Cyma Reversa, the fillet A, the point C, 
the end of the curve B, and the line B. D being given. 

1st. From C, draw C. D, and from J5, draw B. E 
perpendicular to B. D, then draw C, E parallel to B. 
D, which completes the rectangle. 

2nd. Divide the rectangle B. E. C. D into four equal 
parts, by drawing F. G and 0. P. 

3rd. Find the points in the curve as in figs. 3 and 4. 

Note 1.— If we turn the figure over so as to bring the line F. 
G vertical, G being at the top, the point B of fig. 5, to coincide 
with the point A of fig. 3, it will be perceived that the curves are 
similar, P. G being the transverse axis, and JV. if or M, if the 
conjugate axis of the ellipsis. 

Note 2.— 'The nearer the line B. D approaches to a horizontal 
position, the greater will be the degree of curvature, the conjugate 
axis of the ellipsis will be lengthened, and the curve become more 
Uke the Roman ogee. 



PLATE XXVIII. 



87 



Figure 6. — The Grecian Scotia. 



To describe the Grecian Scotia, the position of the 
fillets A and B being given. 

1st. Join A, By bisect it in C, and through C draw 
D. E parallel to B. G. 

2nd. Make C. D and C. E each equal to the depth 
intended to be given to the scotia ; then A. B will be 
a diameter of an ellipsis, and D. E its conjugate. 

3rd. Through E, draw F. G parallel to A. B. 

4th. Divide A. F and B. G into the same number 
of equal parts, and from the points of division draw 
lines to E, 

5th. Divide A. C and B. C into the same number 
of equal parts, as A. jP, then from D through the points 
of division in A, JS, draw lines to intersect the others, 
which will give points in the curve. 



PLATE XXIX. 
TO DRAW THE TEETH OF WHEELS. 



1st. The LINE of CENTRES is the line A. B. D, fig. 
1, passing through K and C, the centres of the wheel 
and pinion. 

3nd. The proportional or primitive diameter of the 
wheel, is the line A. B; the proportional radius A. K. 
or K, B, The true radii are the distances from the 
centres to the extremities of the teeth. 

3rd. The proportional diameter of the pinion is 
the line B. D ; the proportional radius C J5, 



S8 PLATE XXIX* 

4th. The proportional circles or pitch lines 
are circles described with the proportional radii touch- 
ing each other in B* 

5th. The pitch of a wheel is the distance on the 
pitch circle including a tooth and a space, as E* F or 
G. H, or 0. D, fig. 2. 

6th. The depth of a tooth is the distance from the 
pitch circle to the bottom, as i. if, fig. 1, and the 
height of a tooth is the distance from the pitch circle to 
the top of the tooth, as i. M* 



To draw the Pitch Line of a Pinion to contain a defi- 
nite number of Teeth of the same size as in the given 
wheel K. 

1st. Divide the proportional diameter A. B of the 
given wheel into as many equal parts as the wheel has 
teeth, viz. 16. 

2nd. With a distance equal to one of these parts, 
step off on the line B» D as many steps as the pinion is 
to contain teeth, which will give the proportional di- 
ameter of the pinion ; the diagram <;ontains 8. 

3rd. Draw the pitch circle, and on it with the dis- 
tance E. Fy the pitch, lay off the teeth. 

4th. Sub-divide the pitch for the tooth and space, 
draw the sides of the teeth below the pitch line toward 
the centre, and on the tops of the teeth describe epicy- 
cloids. 

Note. — ^The circumferences of circles are directly as their di- 
ameters; if the diameter of one circle be four times greater than 
another, the circumference will also be four times greater. 

Fig. 2 is another method for drawing the teeth ; ji. 
B is the pitch circle on which the width of the teeth 
and spaces must be laid down. Then with a radius 
D. E or D. Fy equal to a pitch and a fourth, from the 



PJ/jtr zn . 



TEETH OF WHEELS 




16 In 14 I.-. 7 2 11 lO ^y f} 7 a ,-> -1- .-? 2 j /.' I L' /i .] .7 6 1 7 /;, 






^V^-^ 




/ ^ 



/--^ \ 



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n/'::.:^i X-Sr 



PLATE XXIX. 



89 



middle of each tooth on the pitch circle as at JD, de- 
scribe the tops of the teeth E and F, from describe 
the tops of the teeth G and D, and so on for the others. 
The sides of the teeth within the pitch circle may be 
drawn toward the centre, as at F and i?, or from the 
centre 0, with a radius equal to O. Q or 0. P, describe 
the lower part of the teeth G and D. 



PLATE XXX. 
ISO METRICAL DRAWING 



Figure 1 



To draw the Isometrical Cube, 

Let j1 be the centre of the proposed drawing. 

1st. With one foot of the dividers in ^, and any ra- 
dius, describe a circle. 

2nd. Through the centre A, draw a diameter B. C 
parallel to the sides of the paper. 

3rd. With the radius from the points B and Clay 
off the other corners of a hexagon, D. E. F. G. 

4th. Join the points and complete the hexagon. 

5th. From the centre ./3, draw lines to the alternate 
corners of the hexagon, which will complete the figure. 

The Isometrical cube is a hexahedron supposed to 
be viewed at an infinite distance, and in the direction 
of the diagonal of the cube; in the diagram, the eye is 
supposed to be placed opposite the point Jl : if a wire 
be run through the point Jl to the opposite corner of 
the cube, the eye being in the same line, could only 
see the end of the wire, and this would be the case no 



90 



PLATE XXX. 



matter how large the cube, consequently the front top 
corner of the cube and the bottom back corner must be 
represented by a dot, as at the point A, As the cube 
is a solid, the eye from that direction will see three of 
its sides and nine of its twelve edges, and as the dis- 
tance is infinite, all these edges will be of equal length, 
the edges seen are those shewn in fig. 1 by continuous 
lines; three of the edges and three of the sides could 
not be seen, these edges are shewn by dotted lines in 
fig. 1, but if the cube were transparent all the edges 
and sides could be seen. The apparent opposite angles 
in each side are equal, two of them being 120^, and 
the other two 60°; all Ihe opposite boundary lines are 
parallel to each other, and as they are all of equal 
length may be measured by one common scale^ and all 
lines parallel to any of the edges of the cube may be 
measured by the same scale. The lines F. G, A. C 
and E. D represent the vertical edges of the cube, the 
parallelograms A. C. D. E and A, C. F. G, represent 
the vertical yace^ of the cube, and the parallelogram A. 
B. E, F represents the horizontal face of the cube ; 
consequently, vertical as well as horizontal lines and 
surfaces may be delineated by this method and measured 
by the same scale, for this reason the term isometri- 
CAL (equally measurable) has been applied to this style 
of drawing. 



Figure 2 



Is a cube of the same size as fig. 1, shaded to make 
the representation more obvious ; the sides of the small 
cube Aj and the boundary of the square platform on 
which the cube rests, as well as of the joists which 
support the floor of the platform, are all drawn parallel 



P]at/> SO 



amSTBUCTION OF THE ISimETIilCAL CUBE 





/»/. 2 




Y-r" M-niTi, 



HI man k Sons 



Flate. 31 



ISOMETRICAJ. FIG JIBES 



Fixf.l . 




Fill. 2 



.^' 




Fi/f . 3 . 



r 








Fuj.4. 



\ 



jr' Mmifli 



Ilbmm S'^'r. 



PLATE XXX. 



91 



to some of the edges of the cube, and forms a good 
illustration for the learner to practice on a larger scale. 

Note. — A singular optical illusion may be witnessed while 
looking at this diagram, if we keep the eye fixed on the point A, 
and imagine the drawing to represent the interior of a room, the 
point A will appear to recede ; then if we again imagine it to be 
a cube the point will appear to advance, and this rising and falling 
may be continued, as you imagine the angle A to represent a pro- 
jecting corner, or an internal angle. 



PLATE XXXI. 
EXAMPLES IN ISOMETRICAL DRAWING. 



Figs. 1 and 2 are plans of cubes with portions cut 
away. Figs. 3 and 4 are isometrical representations 
of them. 

To draw a part of a regular figure, as in these dia- 
grams, it is better to draw the whole outline in pencil, 
as shewn by the dotted lines, and from the corners lay 
off the indentations. 

The circumscribing cube may be drawn as in fig. 1, 
Plate 30, with a radius equal to the side of the plan, 
or with a triangle having one right angle, one angle of 
60^, and the other angle 30^, as shewn at A. Proceed 
as follows : — 

Let B be the tongue of a square or a straight edge 
applied horizontally across the paper, apply the hy- 
pothenuse of the triangle to the tongue or straight edge, 
as in the diagram, and draw the left hand inclined lines; 
then reverse the triangle and draw the right hand in- 
clined lines ; turn the short side of the triangle against 
the tongue of the square, and the vertical lines may be 
drawn, 



92 



PLATE XXXII. 



This instrument so simplifies isometrical drawing, 
that its application is but little more difficult than the 
drawing of flat geometrical plans or elevations. 



PLATE XXXII. 

EXAMPLES IN ISOMETRICAL DRAWING 
CONTINUED. 



Fig. 



1 is the side, and fig. 2 the end elevation of a 
block pierced through as shewn in fig. 1, and with the 
top chamfered off, as shewn in figs. 1 and 2. 



Figure 3. 



To draw the figure Isometrically, 

1st. Draw the isometrical lines ^. B and C D; 
make ^. B equal to •/?. B fig. 1, and C. D equal to C. 
D fig. 2. 

2nd. From Jl. B and D, draw the vertical lines, and 
make them equal to B. G, fig. 1. 

3rd. Draw K. H and L. I parallel to •/?. By and H. 
I and K. L parallel to C. D, 

4th. Draw the diagonals H. D and J. C, and through 
their intersection draw a vertical line M, G. F. Make 
G. F equal to G. F, fig. 1. 

5th. ^Through G, draw G. JV*, intersecting L. K in 
JY. and from JY draw a vertical line Jf. E. 

6th. Through jP, draw F. J5, intersecting JV*. E in 
E ; then E. F represents the line E. Fm fig. 1. 

7th. From E and Fj lay oiF the distances and P, 
and from and P draw the edges of the chamfer 0. 
jfiC — 0. jL — P. Jff and P. J, which complete the outline. 



Flate^ 3Z 



ISOMETHICJIL FIGURES 



Fi^.l. 



E 



/ .10 



Fig.Z. 



(f w^^ 



I \. 



-.' -*= 




^D 



WMuufie. 



lUvimi 'k Sorts. 



PLATE XXXIT, 



93 



8th, On A. B lay off the opening shewn in fig. 1, 
and from i?, draw a line parallel to C. J). 

Note 1. — All the lines in this figure, except the diagonals and 
edges of the chamfer, can be drawn with the triangle and square, 
as explained in Plate 31. 

Note 2. — All these lines may be measured by the same scale, 
except the incKned edges of the chamfer, which will require a 
different scale. 

Note 3. — The intelligent student will easily perceive from this 
figure, how to draw a house with a hipped roof, placing the 
doors, windows, &c., each in its proper place ; or how to draw 
any other rectangular figure. Inclined lines may always be 
found by a similar process to that we have pursued in drawing 
the edges of the chamfer. 



Figure 4 



Is the elevation of the side of a cube with a large por- 
tion cut out. 



Figure 5 



Is the isometrical drawing of the same, with the top of 
the cube also pierced through. The mode pursued is 
so obvious, that it requires no explanation: it is given 
as an illustration for drawing furniture, or any other 
framed object. It requires but little ingenuity to con- 
vert fig. 5 into the frame of a table or a foot-stool. 



94 



PLATE XXXIII, 



PLATE XXXIII. 
TO DRAW THE ISOMETRICAL CIRCLE. 

Figure 1 



Is the plan of a circle inscribed in a square, with two 
diameters A, B and C D parallel to the sides of the 
square. 



Figure 2 



To draw the Isometrical Representation. 

1st. Draw the isometrical square, M, JST. 0. P, hav- 
ing its opposite angles 120^ and 60^ respectively. 

2nd. Bisect each side and draw Jl. B and C. D. 

3rd. From draw 0. A and 0. D, and from M 
draw M. C and M. B intersecting in Q and R. 

4th. From Q, with the radius Q. Ay describe the arc 
A. C, and from i?, with the same radius, describe the 
arc D. B. 

5th. From 0, with the radius 0. Ay draw the arc 
A, D, and from M, with the same radius, describe C. 
By which completes the oval. 

Note. — An isometrical projection of a circle wouM be an 
ellipsis; but the figure produced by the above method is so sim- 
ple in its construction and approaches so near to an ellipsis, that 
it may be used in most cases, besides its facility of construction, 
its circumference is so nearly equal to the circumference of the 
given circle, that any divisions traced on the one may be trans- 
ferred to the other with sufficient accuracy for all practical pur- 
poses. 



Flate J'.^ 



IS OME TRI CJi L C IK C L E 




Ymuie 



1 



PLATE XXXIII. 



95 



Figure 3, 



To divide the drcumference of the Isometrical Circle 
into any number of equal parts. 

1st, Draw the circle and a square around it as in fig. 
2, the square may touch the circle as in fig. 2, or be 
drawn outside as in fig. 3. 

2nd. From the middle of one of the sides as 0, erect 
0, K perpendicular to E. F^ and make 0. K equal to 
0. E. 

3rd. Draw K. E and K, Fy and from K with any 
radius, describe an arc P. Q, cutting K. E in P, and 
K F in Q. 

4th. Divide the arc P 4 into one-eighth of the num- 
ber of parts required in the whole circumference, and 
from Kj through these divisions, draw lines intersecting 
E, in 1, 2 and 3. 

5th. From the divisions 1, 2 and 3, in E. 0, draw 
lines to the centre P, which will divide the arc E. 
into four equal parts. 

6th. Transfer the divisions on E. from the cor- 
ners E, F, G. i?, and draw lines to the centre P, when 
the concentric curves will be divided into 32 equal 
parts. 

Note 1. — If a plan of a circle divided into any number of 
equal parts be drawn, as that of a cog Avheel, the same measures 
may be transferred to the isometric curve as explained in the note 
to fig. 2, but if the plan be not drawn, the divisions can be made 
as in fig. 3. 

Note 2. — ^The term isometrical projection has been avoided, 
as the projection of a figure would require a smaller scale to be 
used than the scale to which the geometrical plans and elevations 
are drawn, but as the isometrical figure drawn with the same 
scale to Avhich the plans are drawn, is in every respect propor- 
tional to the true projection, and conveys to the eye the same 



96 PLATE XXXIII. 

■I . ■ ' » I * < ■■. I 11 1 I . I I III I 

view of the object, it is manifestly much more convenient for 
practical purposes to draw both to the same scale. 

Note 3. — In Note 2 to fig. 3, Plate 32, allusion has been made 
to inclined lines requiring a different scale from any of the Imes 
used in drawing the isometrical cube : for the mode of drawing 
those scales as well as for the further prosecution of this branch 
of drawing, the student is referred to Jopling's and Sopwith's 
treatise on the subject, as we only propose to give an introduc1lt)n 
to isometrical drawing. Sufficient, however, has been given to 
enable the student to apply it to a very large class of objects, and 
it would extend the size of this work too much, to pursue the 
subject in full. 



PEESPECTIVE. 

PLATE XXXIV. 



The design of the art of perspective is to draw on a 
plane surface the representation of an object or objects, 
so that the representation shall convey to the eye, the 
same image as the objects themselves would do if 
placed in the same relative position. 

To elucidate this definition it will be necessary to 
explain the manner in which the image of external ob- 
jects is conveyed to the eye. 

1st. To enable a person to see any object, it is ne- 
cessary that such object should reflect light. 

2nd. Light reflected from a centre becomes weaker 
in a duplicate ratio of distance from its source, it being 
only one-fourth as intense at double the distance, and 
one-ninth at triple the distance, and so on. 

3rd. A ray of light striking on any plane surface, is 
reflected from that surface in exactly the same angle 



PLATE XXXIV. 



97 



with which it impinges ; thus if a plane surface be 
placed at an angle of 45^, to the direction of rays of 
light, the rays will be reflected at an angle of 45^ in the 
opposite direction. This fact is expressed as follows, 

viz : THE ANGLE OF REFLECTION IS EQUAL TO THE 

ANGLE OF INCIDENCE. This axiom, so short and 
pithy, should be stored in the memory with some others 
that we propose to give, to be brought forward and ap- 
plied whenever required. 

4th. Rays of light reflected from a body proceed in 
straight lines until interrupted by meeting with other 
bodies, which by reflection or refraction, change their 
direction. 

5th. Refraction of light. When a ray of light 
passes from a rare to a more dense medium, as from a 
clear atmosphere through a fog or from the air into 
water, it is bent out of its direct course : thus if we 
thrust a rod into water, it appears broken or bent at the 
surface of the water; objects have been seen through a 
fog by the bending of the rays, that could not possibly 
be seen in clear weather ; this bending of the rays of 
light is called refraction, and the rays are said to be re- 
fracted : this effect, (produced however by a different 
cause) may often be seen by looking through common 
window glass, when in consequence of the irregulari- 
ties of its surface, the rays are bent out of a direct course, 
causing the view of objects without to be much distorted. 

6th. A portion of light is absorbed by all bodies re- 
ceiving it on their surface, consequently the amount of 
light reflected from an object is not equal to the quan- 
tity received. 

7th. The amount of absorption is not the same in all 
bodies, but depends on the color and quality of the re- 
flecting surface ; if a ray falls on the bright polished 
surface of a looking-glass, most of it will be reflected, 



98 PLATE XXXIV. 

but if it should fall on a surface of black cloth, most of 
it would be absorbed. White or light colors reflect 
more of a given ray of light than dark colors; polished 
surfaces reflect more than those which are unpolished, 
and smooth surfaces more than rough. 

8th. As all objects absorb more or less light, it fol- 
lows that at each reflection the ray will become weaker 
until it is no longer perceptible. 

9th. Rays received from a luminous source are called 
direct, and the parts of an object receiving these direct 
rays are said to be in light. The portions of the sur- 
face so situated as not to receive the direct rays are 
said to be in shade ; if the object receiving the direct 
rays is opaque, it will prevent the rays from passing in 
that direction, and the outline of its illuminated parts 
will be projected on the nearest adjoining surface : the 
figure so projected is called its shadow. 

10th. The parts of an object in shade will always be 
lighter than the shadow, as the object receives more or 
less reflected light from the atmosphere and adjoining 
objects, the quantity depending on the position of the 
shaded surface, and on the position and quality of the 
surrounding objects. 

11th. If an object were so situated as to receive only 
a direct ray of light, without receiving reflected light 
from other sources, the illuminated portion could alone 
be seen ; but for this universal law of reflection we 
should be able to see nothing that is not illuminated by 
the direct rays of the sun or by some artificial means. 

12th, Rays of light proceeding in straight lines from 
the surfaces of objects, meet in the front of the eye of 
the spectator where they cross each other, and form an 
inverted image on the back of the eye, of all objects 
within the scope of vision. 

13th. The size of the image so formed on the retina 



- "^ 



-I 



i 



i 



Flate 34 



PERSPECTIVE 




yr:' Mim:n. 



IiiniJii.\S, 



PLATE XXXIV. 



99 



depends on the size and distance of the original ; the 
shape of the image depends on the angle at which it is 
seen. 

Note. — ^The size of objects diminishes directly as the distance 
increases, appearing at ten times the distance, only a tenth part 
as large; the knowledge of this fact has produced a system of 
arithmetical perspective, which enables the draughtsman to pro- 
portion the sizes of objects by calculation. 

14th. The strength of the image depends on the de- 
gree of illumination of the original, and on its distance 
from the eye, objects becoming more dim as they re- 
cede from the spectator. 

15th. To give a better idea of the operation of the 
eye in viewing an object, let us refer to fig. 1. The 
circle Jl is intended to represent a section of the human 
eye, H the pupil in front, K the crystalline lens in 
which the rays are all converged and cross each other, 
and M the concave surface of the back of the eye called 
the retina^ on which the image is projected. 

16th. Let us suppose the eye to be viewing the cross 
B, C, and that the parallelogram JV. 0. P. Q represents 
a picture frame in which a pane of glass is inserted; 
the surface of the glass slightly obscured so as to allow 
objects to be traced on it, then rays from every part of 
the cross will proceed in straight lines to the eye, and 
form the inverted image C. B on the retina. If with a 
pencil we were to trace the form of the cross on the 
glass so as to interrupt the view of the original object, 
we should have a true perspective representation of the 
original, which would form exactly the same sized 
image on the retina; thus the point h would intercept 
the view of JB, c of C, d of D and e of £?, and if colored 
the same as the original, the image formed from it 
would be the same in every respect as from the original, 

17th. If we move the cross B, Cio F. G, the image 



100 



PLATE XXXIV. 



formed on the retina would be much larger, as shewn 
at G. Fy and the representation on the glass would be 
larger, the ray from F passing through y*, and the ray 
from G passing through g, shewing that the same ob- 
ject will produce a larger or smaller image on the re- 
tina cts it advances to or recedes from the spectator ; 
the farther it recedes, the smaller will be the image 
formed, until it becomes so small as to be invisible. 

18th. Fig. 2 is given to elucidate the same subject. 
If we suppose a person to be seated in a room, the 
ground outside to be on a level with the bottom of the 
window ^. jB, the eye at S in the same level line, and 
a series of rods C. D. E. F of the same height of the 
window to be planted outside, the window to be filled 
with four lights of glass of equal size, then the ray from 
the bottom of all the rods would pass through the bot- 
tom of the window; the ray from the top of C would 
pass through the top of the window ; from the top of D 
a little farther off, it would pass through the third light; 
the ray from E would pass through the middle, and F 
would only occupy the height of one pane. 

19th. Fig. 3. Different sized and shaped objects 
may produce the same image ; thus the bent rods ^ 
and C, and the straight rods B and D would produce 
the same image, being placed at different distances 
from the eye, and all contained in the same angle D. 
Sf J5. As the bent rods ^ and C are viewed edgewise 
they would form the same shaped image as if they were 
straight, The angle formed by the rays of light pass- 
ing from the top and bottom of an object to the eye, 
as D, S. Ey is called the visual angle, and the 
object is said to subtend an angle of so many degrees, 
measuring the angle formed at S. 

20th. Of FORESHORTENING. When an object is 
viewed obliquely it appears much shorter than if its 



PLATE XXXIV. 



101 



side is directly in front of the eye ; if for instance we 
hold a pencil sidewise at arms length opposite the eye, 
we should see its entire length ; then if we incline the 
pencil a little, the side will appear shorter, and one of 
the ends can also be seen, and the more the pencil 
is inclined the smaller will be the angle subtended by 
its side, until nothing but the end would be visible. 
Again if a wheel be placed perpendicularly opposite the 
eye, its rim and hub would shew perfect circles, and 
the spokes would all appear to be of the same length, 
but if we incline the wheel a little, the circles will ap- 
pear to be ellipses, and the spokes appear of different 
lengths, dependant on the angle at which they are 
viewed ; the more the wheel is inclined the shorter will 
be the conjugate diameter of the ellipsis, until the whole 
would form a straight line whose length would be equal 
to the diameter, and its breadth equal to the thickness 
of the wheel. This decrease of the angle subtended 
by an object, when viewed obliquely, is called fore- 
shortening* 



PLATE XXXV 



Figure 1 



21st. If we suppose a person to be standing on level 
ground, with his eye at S^ the line A. F parallel to the 
surface and about five feet above it, and the surface G. 
E to be divided off into spaces of five feet, as at B. C. 
D and JC, then if from Sy with a radius S. G, we de- 
scribe the arc Ji. G, and from the points B. C. D and 
E we draw lines to Sj cutting the arc in H, K. L and 
My the distances between the lines on the arc, will rep- 



102 



PLATE XXXV, 



resent the angle subtended in the eye by each space, 
and if we adopt the usual mode for measuring an angle, 
and divide the quadrant into 90°, it will be perceived 
that the jGirst space of five feet subtends an angle of 45°, 
equal to one-half of the angle that would be subtended 
by a plane that would extend to the extreme limits of 
vision ; the next space from B to C subtends an angle 
of about 18|^°, from C to D about 8°, and from D to 
E about 4^°, and the angle subtended would constantly 
become less, until the divisions of the spaces would at 
a short distance appear to touch each other, a space of 
five feet subtending an angle so small, that the eye 
could not appreciate it. It is this foreshortening that 
enables us in some measure to judge of distance. 

22nd. If instead of a level plane, the person at S be 
standing at the foot of a hill, the surface being less in- 
clined would diminish less rapidly, but if on the con- 
trary he be standing on the brow of a hill looking down- 
ward, it would diminish more rapidly ; hence we de- 
rive the following axiom: The degree of fore- 
shortening OF OBJECTS depends ON THE ANGLE AT 
WHICH THEY ARE VIEWED. 

23rd. Perspective may be divided into two 
branches, linear and aerial. 

24th. Linear perspective teaches the mode of 
drawing the lines of a picture so as to convey to the 
eye the apparent shape or figure of each object from 
the point at which it is viewed. 

25th. Aerial perspective teaches the mode of 
arranging the direct and reflected lights, shades and 
SHADOWS of a picture, so as to give to each part its 
requisite degree of tone and color, diminishing the 
strength of each tint as the objects recede, until in the 
extreme distance, the whole assumes a bluish gray 
which is the color of the atmosphere. This branch of 



D 



Flate 35 



PERSPE C TIVE . 



li^.l. 




^s 1 



s 







R W ,T 



\ ^ / 



jy' 



W'"M:iaJie. 



'ZiiidJti Sc 



PLATE XXXV. 



103 



the art is requisite to the artist who would paint a land- 
scape, and can be better learnt by the study of nature 
and the paintings of good masters, than by any series 
of rules which would require to be constantly varied. 

26th. Linear perspective, on the contrary, is capable 
of strict mathematical demonstration, and its rules must 
be positively followed to produce the true figure of an 
object. 

DEFINITIONS. 



27th, The perspective plane is the surface on 
w^hich the picture is drawn, and is supposed to be 
placed in a vertical position between the spectator and 
the object — thus in fig. 1, Plate 34, the parallelogram 
JY, 0. P. Q is the perspective plane. 

28th. The ground line or base line of a picture 
is the seat of the perspectiv^e plane, as the line Q. P, 
fig. 1, Plate 34, and G. L, fig. 2, Plate 35. 

29th. The Horizon. The natural horizon is the 
line in which the earth and sky, or sea and sky appear 
to meet; the horizon in a perspective drawing is at 
the height of the eye of the spectator. Ifihe object 
viewed be on level ground, the horizon will be about 
five feet or five and a half feet above the ground line, 
as it is represented by F. i, fig. 2. If the spectator 
be viewing the object from an eminence, the horizon 
will be higher, and if the spectator be low^er than the 
ground on which the object stands, the horizon will be 
lower ; thus the horizon in perspective, means the 
height of the eye of the spectator, and as an object may 
be viewed by a person reclining on the ground, or 
standing upright on the ground, or he may be elevated 
on a chair or table, it follows that the horizon may be 
made higher or lower, at the pleasure of the draughts- 



104 PLATE XXXV. 

man ; but in a mechanical or afchitectural view of a 
design, it should be placed about five feet above the 
ground line. 

Note. — The tops of all horizontal objects that are below the 
horizon, and the under sides of objects above the horizon, will 
appear more or less displayed as they recede from or approach to 
the horizon. 

30th. The station point, or point of view is the 
position of the spectator when viewing the object or 
picture. 

31st. The point of sight. If the spectator stand- 
ing at the station point should hold his pencil horizon- 
tally at the level of his eye in such a position that the 
end only could be seen, it would cover a small part of 
the object situated in the horizon ; this point is marked 
as at Sy fig. 2, and called the point of sight. It must 
be remembered that the point of sight is not the position 
of a spectator when vieiving an object; but a point in 
the horizon directly opposite the eye of the spectator, 
and from which point the spectator may be at a greater 
or less distance. 

32nd. Points of distance are set off on the horizon 
on either side of the point of sight as at D. D', and 
represent the distance of the spectator from the per- 
spective plane. As an object may be viewed at dif- 
ferent distances from the perspective plane, it follows 
that these points may be placed at any distance from 
the point of sight to suit the judgment of the draughts- 
man, but they should never be less than the base of the 
picture. 

Note 1. — Although the height of the horizon, and the points 
of distance may be varied at pleasure, it is only from that distance 
and with the eye on a level with the horizon that a picture can 
be viewed correctly. 

Note 2. — In the following diagrams the points of distance 
have generally been placed within the boundary of the plates, as 



PLATE XXXV. 



105 



it is important that the learner should see the points to which 
the lines tend ; they should he copied with the points of distdntt 
much farther off, 

33rd. Visual Rays. All lines drawn from the ob* 
ject to the eye of the spectator are called visual rays* 

34th. The middle ray, or central visual ray is 
a line proceeding from the eye of the spectator to the 
point of sight ; external visual rays are the rays pro* 
ceeding from the opposite sides of an object, or from 
the top and bottom of an object to the eye. The angle 
formed in the eye by the external rays, is called the 
visual angle. 

Note. — ^The perspective plane must always be perpendicular 
to the middle visual ray. 

35th. Vanishing Points. It has been shewn at 
fig. 1 in this plate that objects of the same size subtend 
a constantly decreasing angle in the eye as they recede 
from the spectator, until they are no longer visible ; 
the point where level objects become invisible or appear 
to vanish, will always be in the horizon, and is called 
the vanishing point of that object. 

36th. The point of sight is called the principal 
VANISHING point, becausc all horizontal objects that 
are parallel to the middle visual ray will vanish in that 
point. If we stand in the middle of a street looking 
directly toward its opposite end as in Plate 46, (the 
Frontispiece^) all horizontal lines, such as the tops and 
bottoms of the doors and windows, eaves and cornices 
of the houses, tops of chimnies, &c. will tend toward 
that point to which the eye is directed, and if the lines 
were continued they would unite in that point. Again, 
if we stand in the middle of a room looking towards its 
opposite end, the joints of the floor, corners of ceiling, 
washboards and the sides of furniture ranged against 
the side walls, or placed parallel to them, would all tend 
to a point in the end of the room at the height of the eye. 



106 



PLATE XXXV. 



37th. The vanishing points of horizontal objects 
not parallel with the middle ray will be in some point 
of the horizon, but not in the point of sight* These 
vanishing points are called accidental points,, 

38th. Diagonals. Lines drawn from the perspec- 
tive plane to the point of distance as JV. D' and 0. D, 
or from a ray drawn to the point of sight as E, D' and 
jP. Dy are called diagonals ; all such lines represent 
lines drawn at an angle of 45^ to the perspective plane, 
and form as in this figure the diagonals of a square, 
whose side is parallel to the perspective plane* 

39th» Of vanishing PLANES. On taking a position 
in the middle of a street as described in paragraph 36, 
it is there stated that all lines will tend to a point in 
the distance at the height of the eye, called the point 
of sight, or principal vanishing point ; this is equally 
true o( horizontal or vertical planes that are parallel to the 
middle visual ray : for if we suppose the street between 
the curb stones, and the side walks of the street to be 
three parallel horizontal planes as in Plate 46, their 
boundaries will all tend to the vanishing point, until at 
a distance, depending on the breadth of the plane, they 
become invisible. Again, the walls of the houses on 
both sides of the street are vertical planes, bounded by 
the eaves of the roofs and by their intersection with the 
horizontal planes of the side walks, these boundaries 
would also tend to the same point, and if the rows of 
houses, were continued to a sufficient distance, these 
planes would vanish in the same point; if the back 
walls of the houses are parallel to the front, the planes 
formed by them will vanish in the same point, and if 
any other streets should be running parallel to the first, 
their horizontal and vertical planes would all tend to 
the same point. 

Note. — A bird's eye view of the streets of a town laid out 



PLATE XXXV. 



107 



regularly, would fully elucidate the truth of the remarks in this 
paragraph. When the horizon of a picture is placed very high 
above the tops of the houses, as if the spectator were placed on 
some very elevated object, or if seen as a bird would see it when 
on the wing, the view is called a bird's eye view; in a representa- 
tion of this kind the tops of all objects are visible, and the tenden- 
cy of all the planes and lines parallel to the middle visual ray to 
vanish in the point of sight, is very obvious. 

40th. If we were viewing a room as described in 
paragraph 36, the ceiling and floor would be horizontal 
planes, and the walls vertical planes, and if extended 
would all vanish in the point of sight ; or if we were 
viewing the section of a house of several stories in 
height, all the floors and ceilings would be horizontal 
planes, and all the parallel partitions and walls would 
be vertical planes, and would all vanish in the same 
point. 

41st. When the boundaries of inclined planes 
are horizontal lines parallel to the middle ray, the planes 
will vanish in the point of sight ; thus the roofs of the 
houses in Plate 46, bounded by the horizontal lines of 
the eaves and ridge, are inclined planes vanishing in 
the point of sight. 

42nd. Planes parallel to the plane of the 
PICTURE have no vanishing point, neither have any 
lines drawn on such planes. 

43rd. Vertical or horizontal parallel planes 
running at any inclination to the middle ray or perspec- 
tive plane, vanish in accidental points in the horizon, 
as stated in paragraph 37; as for example, the walls 
and bed of a street running diagonally to the plane of 
the picture, or of a single house as in Plate 45, where 
the opposite sides vanish in accidental points at differ- 
ent distances from the point of sight, because the walls 
form different angles with the perspective plane, as 
shewn by the plan of the walls J5. D and D. C, fig. 1. 



108 



PLATE XXXV, 



44th. All horizontal lines drawn on a plane, 
or running parallel to a plane, vanish in the same point 
as the plane itself. 

45th. Inclined lines vanish in points perpendicu- 
larly above or below the vanishing point of the plane, 
and if they form the same angle with the horizon in 
different directions as the gables of the house in fig. 2, 
Plate 45, the vanishing points will be equidistant from 
the horizon. 

From what has been said we derive the following 
axioms ; their importance should induce the student to 
fix them well in his memory : 

1st. The ANGLE OF REFLECTION OF LIGHT is eqUal 

to the angle of incidence. See paragraph No. 3, page 
96. 

2nd. The shadow of an object is always darker 
than the object itself. See paragraph 10, page 98. 

3rd. The degree of foreshortening of objects 
depends on the angle at which they are viewed. See 
paragraph 20, page 100. 

4th. The apparent size of an object decreases ex- 
actly as its distance from the spectator is increased. — 
See paragraph 35, p. 105. 

5th. Parallel planes and lines vanish to a 
common point. See paragraph 36, page 105. 

6th. All parallel planes whose boundaries are 
parallel to the middle visual ray, vanish in the point 
of sight. See paragraph 36, page 105. 

7th. All horizontal lines parallel to the middle 
ray vanish in the point of sight. 

8th. Horizontal lines at an angle of 45° with 
the plane of the picture, vanish in the pokits of dis- 
tance. See paragraph 38, page 106. 

9th. Planes and lines parallel to the plane 
OF the picture have no vanishing point. 



PLATE XXXV. 



109 



PRACTICAL PROBLEMS 



1st. To draw the perspective representation of the 
square N. O. P. Q, viewed in the direction of the line 
W. B, with one of its sides N. touching the perspec- 
tive plane G. L, and parallel with it, 

1st. Draw the horizontal line V. L at the height of 
the eye. 

2nd. From C, the centre of the side JV. 0, draw a 
perpendicular to F. L, cutting it in S» Then S is the 
point of sight or the principal vanishing point, and C- 
S the middle visual ray. 

3rd. As the sides JV. P and 0. Q are parallel to the 
middle ray C. S^ they will vanish in the point of sight. 
Therefore from JV and draw rays to S ; these are 
the external visual rays. 

4th. From S^ set off the points of distance D. D' at 
pleasure, equidistant from S^ and from JV and 0, draw 
the diagonals JV. jy and 0. D, Then the intersection 
of these diagonals with the external visual rays deter- 
mine the depth of the square. 

5th. Draw E. F parallel to JV. 0. Then the trape- 
zoid JY. 0. E. F is the perspective representation of 
the given square viewed at a distance from W on the 
line W. J5, equal to S. D. 

2nd. To draw the Representation of another Square 
of the same size immediately in the rear o/^E. F. 
1st. From jB, draw E. D', intersecting 0. S in H, 

and from F, draw F. D, intersecting JY. S in B. 

2nd. Draw B. H parallel to E, JP, which completes 

the second square ; and the trapezoid JY. 0. H. B is 

the representation of a parallelogram whose side 0. H 

is double the side of the given square. 



110 PLATE XXXV. 

Note. — If from IV on the line JV» B we set ojfT the distance 
S, D, extending in the example outside of the plate^ (which rep- 
resents the distance from which the picture is viewed,) and from 
JV and O draw rays to the point so set off, cutting P. Q in il 
and T^ then the lines U. T and E. F will be of equal length, and 
prove the correctness of the diagram. 



PLATE XXXVI 

Figure 1. 



To draw a Perspective Plan of a Square and divide it 
into a given number of Squares^ say sixty-four. 

Let G. L be the base line, F. L the horizon, *Sthe 
point of sight, and JV. the given side of the square. 

1st. From JV* and 0, draw rays to S and diagonals 
to jD. J), intersecting each other in P and Q, draw 

p. Q. 

2nd. Divide JV. into eight equal parts, and from 
the points of division draw rays to S. 

3rd. Through the points of intersection formed with 
those rays by the diagonals, draw lines parallel to JV*. 
0, which will divide the square as required, and may 
represent a checker board or a pavement of square 
tiles. 

Of Half Distance. 



When the points of distance are too far off to be used 
conveniently, half the distance may be used ; as for ex- 
ample, if we bisect S. D in ^ D, and JVl in C, and 
draw a line from C to ^ D, it will intersect JV. S in P, 
being in the same point as by the diagonal drawn from 



Plate. 36 



PE K S FE C TI VE 




M 



M - 



Mimfu 



'iinicin 7c .J ens. 



PLATE XXXVI. 



Ill 



the opposite side of the square, to the whole distance 
at jD. 

Note. — Any other fraction of the distance may be used, pro- 
vided that the divisions on the base Hne be measured proportion- 
ately. 



Figure 2. 



To draw the Plan of a Room with Pilasters at its sides^ 
the base line^ horizon^ point of sights and points of 
distance given. 

Note. — ^To Avoin repetitions in the following diagranns, we 
shall suppose the base line, the horizon V. L, the point of sight S, and 
the points of distance D,D to be given, 

1st. Let JV. be the width of the proposed room, 
then draw JY, S and 0. S representing the sides of the 
room. 

2nd. From JV toward lay down the width of each 
pilaster, and the spaces between them, and draw lines 
to D, then through the points where these lines inter- 
sect the external visual ray JY. S^ draw lines parallel 
with JY. to the line 0. S. 

3rd. From JY and 0, set off the projection of the 
pilasters and draw rays to the point of sight. The 
shaded parts shew the position of the pilasters. 

4th. If from JY we lay off the distances and widths 
of the pilasters toward Jkf, and draw diagonals to the 
opposite point of distance, JY. S would be intersected 
in exactly the same points. 

Note. — ^Any rectangular object may be put in perspective by 
this method, without the necessity of drawing a geometrical plan, 
as the dimensions may all be laid off on the ground line by any 
scale of equal parts. 



112 PLATE XXXVI, 



Figure 3. 



To shorten the depth of a perspective drawings thereby 
producing the same effect as if the points of dis* 
tance were removed much farther off. 

1st. Let all the principal lines be given as above, 
and the pilasters and spaces laid off on the base line 
from JY. 

2nd. From the dimensions on the base line draw 
diagonals to the point of distance D. The diagonal 
from M the outside pilaster will intersect JV*. S in P. 

3rd. From JY erect a perpendicular JV. B to inter- 
sect the diagonals, and from those intersections draw 
horizontal lines to intersect JV. S. 

4th. If from JY we draw the inclined line JY. E and 
transfer the intersections from it to F. 0, it will reduce 
the depth much more, as shown at 0. S* 

Most of the foregoing diagrams may be drawn as 
well with one point of distance as with two. 



PLATE XXXVII. 
TESSELATED PAVEMENTS 



Figure 1 



To draw a pavement of square tiles^ with their sides 
placed diagonally to the perspective plane. 

1st. Draw the perspective square JY, 0. P. Q. 

2nd. Divide the base line JY. into spaces equal to 
the diagonal of the tiles. 

3rd. From the divisions on JY. draw diagonals to 
the points of distance, and from the intersections of the 



riate. 37. 



PER SPE C TIKE 

Tesselated/ PavejnerUs 




]^ e f a i 2 b (- do 



R"''" J///7//7/ 



:.Un:..zx: IrScK 



PLATE XXXVII. 



113 



diagonals with the external rays draw other diagonals 
to the opposite points of distance. 

4th. Tint every alternate square to complete the 
diagram. 

Figure 2. 

To draw a pavement of square black tiles with a white 
border around them, the sides of the squares parallel 
to the perspective plane and middle visual ray. 

1st. Draw the perspective square, and divide X 
into alternate spaces equal to the breadth of the square 
and borders. 

2nd. From the divisions on X draw rays to 
the point of sight, and from X draw a diagonal to the 
point of distance. 

3rd. Through the intersections formed by the diago- 
nal, with the rays drawn from the divisions on X. 0, 
draw lines parallel to X 0, to complete the small 
squares. 

Figure 3. 



To draw a Pavement composed of Hexagonal and 
Square Blocks, 

1st. Diviae the diameter of one of the proposed 
hexagons a. b into three equal parts, and from the 
points of division draw rays to the point of sight. 

2nd. From a, draw a diagonal to the point of dis- 
tance, and through the intersections draw the parallel 
lines. 

3rd. From 1, 2, 3 and 4, draw diagonals to the 
opposite points of distance, which complete the hexa- 



gon. 



10'' 



114 PLATE XXX VII. 

4th. Lay off the base line from a and b into spaces 
equal to one-third of the given hexagon, and draw rays 
from them to the point of sight ; then draw diagonals 
as in the diagram, to complete the pavement. 



PLATE XXXVIII. 

Figure L 



To draw the Double Square E. F. G. H, viewed diago- 
nally^ with one of its corners touching the Perspective 
Plane, 

1st. Prolong the sides of the squares as shewn by 
the dotted lines to intersect the perspective plane. 

2nd. From the points of intersection, draw diagonals 
to the points of distance, their intersections form the 
diagonal squares. 

3rd. The square A. B. JV. is drawn around it on 
the plan and also in perspective, to prove that the same 
depth and breadth is given to objects by both methods 
of projection. 

Figure 2. 



To draw the Perspective Representation of a Circle 
viewed directly in front and touching the Perspective 
Plane. 

Find the position of any number of points in the Curve. 
1st. Circumscribe the circle with a square, draw the 
diagonals of the square P. and JY. Q, and the di- 
ameters of the circle A. B and E. F, also through the 



D 



Plate 38. 



PERSPECTIVE 



tli).l 



D 




W": Mirahf^. 



IJlrnj^n k. Snv,s 



PLATE XXXVIII. 



115 



intersections of said diagonals with the circumference, 
draw the chords R. R, R, jR, continued to meet the 
line G. Lin Fand YJ 

2nd. Put the square in perspective as before shewn, 
draw the diagonals JV. D', and 0. D, and the radials 
F. S and YJ S. 

3rd. From ^^ draw Ji, S\ and through the intersec- 
tion of the diagonals draw E. F parallel to JY. 0. 

4th. Through the points of intersection thus found, 
viz: ^. B. E. F, R. R. R. R trace the curve. 

Note 1. — ^This method gives eight points through which to 
trace the curve, and as these points are equidistant in the plan, it 
follows that if the points were joined by right lines it would give 
the perspective representation of an octagon. By drawing di- 
ameters midway between those already drawn on the plan, eight 
other points in the curve may be found. This would give six- 
teen points in the curve, and render the operation of tracing much 
more correct. 

Note 2. — A circle in perspective may be considered as a po- 
lygon of an infinite number of sides, or as a figure composed of 
an infinite number of points, and as any point in the curve may 
be found, it follows that every point may be found, and each be 
positively designated by an intersection ; in practice of course this 
is unnecessary, but the student should remember, that the more 
points he can positively designate without confusion, the more 
correct will be the representation. 



PLATE XXXIX. 
LINE OF ELEVATION 



Figure 2 



Is the plan of a square w^hose side is nine feet, each 
side is divided into nine parts, and lines from the divi- 



116 



PLATE XXXIX, 



sions drawn across in opposite directions ; the surface 
is therefore divided into eighty-one squares. G. Ly 
fig. 1, is the base line and D. D the horizon. 

1st.— -7b put the plan with its divisions in perspective y 
one of its sides N. to coincide with the perspective 
plane. 

Transfer the measures from the side JY. 0, fig. 2, to 
to JV. on the perspective plane fig. 1, and put the 
plan in perspective by the methods before described. 

2nd, — To erect square pillars on the squares N. Q. W, 
7iine feet high and one foot diameter ^ equal to the 
size of one of the squares on the plan. 

1st. Erect indefinite perpendiculars from the corners 
of the squares. 

2nd. On JV. A one of the perpendiculars that touches 
the perspective plane lay off the height of the column 
JV. M from the accompanying scale, then JV. M is a 
LINE OF HEIGHTS ou which the true measures of the 
heights of all objects must be set. 

3rd. Two lines drawn from the top and bottom of an 
object on the line of heights to the point of sight, point 
of distance, or to any other point in the horizon, forms 
a scale for determining similar heights on any part of 
the perspective plan. To avoid confusion they are 
here drawn to the point j5. 

4th. Through M draw M. C parallel to JV. 0, and 
from C draw a line to the point of sight which deter- 
mines the height of the side of the column, and also of 
the back column erected on Q, and through the inter- 
section of the line C. S with the front perpendicular, 
draw a horizontal line forming the top of the front side 
of the column Q. 



rUitp 39 



LINE OF ELEVATION 



D B/ 



Fijfj.l. 




D 



X F 



Fifj.Z. 



Q 



^ 














1 


^ 


' 1 














-■■^ 












































,w 




















r 
































/' 










■/■ 










^^H5 



3 ^ -i -^ 1 O 



■Sralc of Fivt 



VlP'^ Wriri 



rihr^Ui- bcSoii 



PLATE XXXIX, 



117 



5th. To determine the height of the pillar at TF, 1st. 
draw a horizontal line from its foot intersecting the 
proportional scale JV. B in Y; 2nd. from Y draw a 
vertical line intersecting M. B in X; then Y, Xis the 
height of the front of the column W. By the same 
method the height of the column Q may be determined 
as shewn at R. T, 

3rd. — To draw the Caps on the Pillars. 

1st. On the line C. E 2i continuation of the top of 
the front, set off the amount of projection C, Ey and 
through E draw a ray to the point of sight. 

2nd. Through C draw a diagonal to the point of 
distance, and through the point of intersection of the 
diagonal with the ray last drawn, draw the horizontal 
line informing the lower edge of the front of the cap. 

3rd. Through Jlfdrawa diagonal to the opposite 
point of distance, which determines the position of the 
corners H and iC, from H draw a ray to the point of 
sight. 

4th. Erect perpendiculars on all the corners, lay off 
the height of the front, and draw the top parallel with 
the bottom. A ray from the corner to the point of 
sight, will complete the cap. 

The other caps can be drawn by similar means. 

As a pillar is a square co/wmn, the terms are here 
used indiscriminately. 

4th. — To erect Square Pyramids on O and P of the 
same height as the Pillars^ with a base of four square 
feet J as shewn in the plan. 

1st. Draw diagonals to the plan of the base, and 
from their intersection at R draw the perpendicular 
R.' T. 



118 PLATE XXXIX. 

2nd. From B! draw a line to the proportional scale 
Jf. Bj and draw the vertical line Z. G, which is the 
height of the pyramid. 

3rd. Make RJ T equal to Z. G, and from the cor- 
ners of the perspective plan draw lines to T', which 
complete the front pyramid. 

4th. A line drawn from T to the point of sight will 
determine the height of the pyramid at a. 

Note 1. — ^The point of sight S shewn in front of the column 
Wy must be supposed to be really a long distance behind it, but 
as we only see the end of a line proceeding from the eye to the 
point of sight, we can only represent it by a dot. 

Note 2. — A part of the front column has been omitted for the 
purpose of shewing the perspective sections of the remaining 
parts, the sides of these sections are drawn toward the point of 
sight, the front and back lines are horizontal. The upper section 
is a little farther removed from the horizon, and is consequently a 
little wider than the lower section. This may be taken as an il- 
lustration of the note to paragraph 29 on page 104, to which the 
reader is referred. 

Note 3. — The dotted lines on the plan shew the direction and 
boundaries of the shadows ; they have been projected at an angle 
of 45^ with the plane of the picture. 



PLATE XL 

Figure 1 



To draw a Series of Semicircular Arches viewed direct- 
ly in fronts forming a Vaulted Passage^ tvith pro- 
jecting libs at intervals J as shewn by the tinted plan 
below the ground line. 

1st. From the top of the side walls JV. /and 0. K, 
draw the front arch from the centre Hj and radiate the 
joints to its centre. 



FLat£40. 
ARCHES IN VERSPECTni^ 



Fuj.l 










^l \ fj 



s/l| 



L-iP- 



K 



t> 



B ■~^. 



A" 



^M. 



- ■i^^^^ r;s^ ^g 



m^ 



Bl- E 



F i- 



I 




11/^' 1^7«/fe, 



PLATE XL. 



119 



2nd. From the centre H and the springing lines of 
the arch, and from the corners A and M draw rays to 
the point of sight. 

3rd. From A and M set off the projection of the ribs, 
and draw rays from the points so set off to the point of 
sight. 

4th. Transfer the measurements of A^^. W. C, &c. 
on the plan, to AK B\ C, &c., on the ground line, and 
from them draw diagonals to the point of distance, in- 
tersecting the ray A. S in B. C D, &c. 

5th. From the points of intersection in A. S draw 
lines parallel to the base line to intersect M. S. This 
gives the perspective plans of the ribs. 

6th. Erect perpendiculars from the corners of the 
plans to intersect the springing lines, and through these 
intersections draw horizontal dotted lines, then the 
points in which the dotted lines intersect the ray drawn 
from H the centre of the front arch, will be the centres 
for drawing the other arches ; R being the centre for 
describing the front of the first rib. 

7th. The joints in the fronts of the projecting ribs 
radiate to their respective centres, and the joints in 
the soflSt of the arch radiate to the point of sight. 

Note. — ^No attempt is made in this diagram to project the 
shadows, as it would render the lines too obscure. But the front 
of each projection is tinted to make it more conspicuous. 



Figure 2. 



To draw Semicircular or Pointed Arcades on either 
side of the spectator ^ runniiig parallel to the middle 
visual ray, N. P and Q. O the width of the arches 
being given y and P. Q,the space between them. 

1st. From JV. P. Q and erect perpendiculars, 
make them all of equal length, and draw E, i^and M, J. 



120 



PLATE XL, 



2nd. For the semicircular arches, bisect E. F 
in Cy and from E. C. F. and Q, draw rays to the 
point of sight. 

3rd. From C, describe the semicircle E. F. 

4th. Let the arches be the same distance apart as 
the width Q. 0, then from draw a diagonal to the 
point of distance, cutting Q. S in i?, from R draw a 
diagonal to the opposite point of distance cutting 0. 
S in Vy from V draw a diagonal to JD, cutting Q. S in 
TF, and from Wto D', cutting 0. 5' in X 

5th. Through i?. F. TFand X, draw horizontal lines 
to intercept the rays 0. iS and Q. S, and on the inter- 
sections erect perpendiculars to meet the rays drawn 
from E and F. 

6th. Connect the tops of the perpendiculars by hori- 
zontal lines, and from their intersections with the ray 
drawn from C in 1, 2, 3 and 4, describe the retiring- 
arches. 

7th. For the gothic arches, (let them be drawn 
the same distance apart as the semicircular,) continue 
the horizontal lines across from R and F, to intersect 
the rays P. S and JV. S, and from the points of inter- 
section erect perpendiculars to intersect the rays drawn 
from Jlf and /. 

8th. From M and /successively, with a radius M. 
Jy describe the front arch, and from iJ the crown, draw 
a ray to S ; from A and B with the radius •/?. By de- 
scribe the second arch, and from K and Z, describe the 
third arch. 

Note.- — All the arches in this plate are parallel to the plane of 
the picture, and although each succeeding arch is smaller than 
the arch in front of it, all may be described with the compasses. 




]/[r"Mim£p. 



HJjvMn k Sons 



PLATE XLI. 



121 



PLATE XLI. 

TO DESCRIBE ARCHES ON A VANISHING PLANE. 



Figure 1 



The Front Arch A. N. B, the Base Line G. L, Hori- 
zon D, S, Point of Sight S, and Point of Distance D, 
being given, 

1st. Draw H, /across the springing line of the arch, 
and construct the parallelogram E, F. J. H. 

2nd. Draw the diagonals H. F and /. E, and a hori- 
zontal line K. Jkf, through the points where the diago- 
nals intersect the given arch. Then H. K. JV. M and 
Jy are points in the curve which are required to be 
found in each of the lateral arches. 

3rd. From F and JB, draw rays to the point of sight 
S, Then if we suppose the space formed by the tri- 
angle B, S. F to be a plane surface, it will represent 
the vanishing plane on which the arches are to be 
drawn. 

4th. From i?, set off the distance B. A to Z, and 
draw rays from Z. J and C, to the point of sight. 

5th. From Z, draw a diagonal to the point of dis- 
tance, cutting B, S in 0; through 0, draw a horizon- 
tal line cutting Z. S in P ; from P, draw a diagonal 
intersecting JB. 5* in Q; through Q, draw a horizontal 
line, cutting Z. S in it, and so on for as many arches 
as may be required. 

6th From 0. Q. S and f/, erect perpendiculars, 
cutting F. S in F. W, X and Y. 



122 PLATE XLI. 

7th. Draw the diagonals J. V, F. J, &c. as shewn in 
the diagram, and from their intersection erect perpen- 
diculars to meet F. S ; through which point and the 
intersections of the diagonals with C. ^S trace the curves. 



Figure 2, 



To draw Receding Arches on the Vanishing Plane J. 
S. D, with Piers between them^ corresponding with 
the given front view^ the Piers to have a Square Base 
with a side equal to C. D. 

1st. From D on the base line, set off the distances 
D. C, a B and B. A to D. E, E. F and F. G, and 
from E. F. G, &c. draw diagonals to the point of dis- 
tance to intersect D. S, 

2nd. From the intersections in D. S, erect perpen- 
diculars ; draw the parallelogram M. JY, H, I around the 
given front arch, the diagonals M, I and H, JV, and 
the horizontal line L. K, prolong H, I to J and J\L JY 
to V. 

3rd. From B. C. D. M. f. J. K and F, draw rays 
to the point of sight, put the parallelograms and diago- 
nals in perspective at 0, P. V. W and at Q. W. R. X, 
and draw the curves through the points as in the last 
diagram. 

4th. From i where E, D' cuts D. 5^, draw a horizon- 
tal line cutting B. S in A, and from h erect a perpen- 
dicular cutting Jkf. S in k. 

5th. From F, the centre of the front arch, draw a 
ray to the point of sight, and from ky draw a horizontal 
line intersecting it in Z. Then Z is the centre for de- 
scribing the back line of the arch with the distance Z, 
k for a radius. 



PLATE XLI. 



123 



Note. — ^The backs of the side arches are found by the same 
method as the fronts of those arches. The lines are omitted to 
avoid confusion. 

The projecting cap in this diagram is constructed in the same 
manner as the caps of the pillars in Plate 39. 



PLATE XLII. 



APPLICATION OF THE CIRCLE WHEN PARAL- 
LEL TO THE PLANE OF THE PICTURE. 



V, L is the horizon, and S the point of sight. 
Figure 1. 



To draw a Semicircular Thin Band placed above the 

horizon. 

Let the semicircle Jl. B represent the front edge of 
the band, A, B the diameter, and C the centre. 

1st. From Ji, C and J5, draw rays to the point of 
sight. 

2nd. From C the centre, lay off toward J5, the 
breadth of the band C E. 

3rd. From E, draw a diagonal to the point of dis- 
tance, intersecting C. S in F, Then F is the centre for 
describing the back of the band. 

4.th. Through jP, draw a horizontal line intersecting 
A. S in K, and B. S in L. Then F. K or F. L is the 
radius for describing the back of the band. 



124 PLATE xm. 



Figure 2. 



To draw a Circular Hoop with its side resting on the 

Horizon. 

The front circle A. H. B. Ky diameter A. By and 
centre C being given. 

1st. From A. C and B, draw rays to the point of 
sight. 

2nd. From C the centre, lay off the breadth of the 
hoop at E. 

3rd. From Ey draw a diagonal to JD', intersecting 
C. S in jP, and through Fy draw a horizontal line inter- 
secting Ji. S in Ky and B. S in L. 

4th. From F with a radius F. L or F. Ky describe 
the back of the curve. 



Figure 3. 



To draw a Cylindrical Tub placed below the Horizon, 
whose diameter y depth and thickness are given. 

1st. From the centre C describe the concentric cir- 
cles forming the thickness of the tub, lay off the staves 
and radiate them toward C 

2nd. Proceed as in figs. 1 and 2 to draw rays and a 
diagonal to find the point F, and from F describe the 
back circles as before ; the hoop may be drawn from F, 
by extending the compasses a little. 

3rd. Radiate all the lines that form the joints on the 
sides of the tub toward the point of sight. 



Figure 4 



Is a hollow cylinder placed below the horizon, and 



PLATE XLII» 



125 



must be drawn by the same method as the preceding 

figures ; the letters of reference are the same. 

Note. — ^The objects in this Plate are tinted to shew the differ- 
ent surfaces more distinctly, without attempting to project the 
shadows. 



PLATE XLIII. 



The object and point of view given^ to find the Perspec- 
tive Plane and Vanishing Points, 

Rule 1. — The Perspective plane must be drawn 
perpendicular to the middle visual ray. 

Rule 2. — The Vanishing Point of a line or plane 
is found by drawing a line through the station point, 
parallel with such line or plane to intersect the perspec- 
tive plane. The point in the horizon immediately over 
the intersection so found, is the vanishing point of all 
horizontal lines in said plane, or on any plane parallel 
to it. 

1st. Let the parallelogram E, F. G. H be the plan 
of an object to be put in perspective, and let Q be the 
position of the spectator viewing it, (called the point 
of view or station pointy) with the eye directed toward 
Ky then Q. K will be the central visual ray, and K the 
point of sight. Draw F. Q and H. Q, these are the 
external visual rays. 

Note. — ^The student should refer to paragraphs 30 and 31, p. 
104, for the definitions of station point and point of sight, 

2nd. Draw P. at right angles to Q. iC, touching 
the corner of the given object at JE, then P. O will be 
the base of the perspective plane. 



IV 



126 PLATE XLIII. 

Note — ^This position of the perspective plane, is the farthest 
point from the spectator at which it can be placed, as the whole 
of the object viewed must be behind it; but it may be placed at 
any intermediate point nearer the spectator parallel with P. O. 

3rd. Through Q draw Q. P parallel with E. Fy in- 
tersecting the perspective plane in P, then P is the van- 
ishing point of the lines E. F and G. H. 

4th. Through Q draw Q. 0, parallel to E. H^ inter- 
secting the perspective plane in 0, then is the van- 
ishing point for E. H and F. G. 

5th. If we suppose the station point to be removed 
to A^ then A, M will be the central visual ray, A. F 
and A. H the external rays, and B. D the perspective 
plane, B the vanishing point of E. F and G, H^ and 
the vanishing point of E, H and F, G will be outside 
the plate about vii inches distant from Aj in the direc- 
tion o{ A, C. 

6th. If the station point be removed to if, it will be 
perceived that E. II and F, G will have no vanishing 
point, because they are perpendicular to the middle ray, 
and a line drawn through the station point parallel with 
the side E. H will also be parallel with the perspective 
plane, consequently could never intersect it. 

7th. The sides E. F and G. Hot the plan, would 

vanish in the point of sight, but if an elevation be 

drawn on the plan in that position which should extend 

above the horizon, then neither of those sides could be 

seen, and the drawing would very nearly approach to a 

geometrical elevation of the same object. 

Note. — In the explanation of this plate, the intersections giv- 
ing the point of sight and vanishing points, are made in the per- 
spective plane, which the student will remember when used in 
this connection, is equivalent to the base line or ground line of 
the picture, being the seat or position of the plane on which the 
drawing is to he made; but we must suppose these points to be 
elevated to the height of the eye of the spectator; in practice, these 



Flate 43 



THE OBJECTAJS^D POINT OF VIEW GIVEN 
TO FIND THE PERSPECTIVE PLANE 
AND JANISHING POINTS . 




imnan he o oris. 



PLATE XLIV. 



127 



points must be set off on the horizontal line as described in para- 
graph 32, page 104. 



PLATE XLIV. 



To delineate the perspective appearance of a Cube view- 
ed accidentally and situated beyond the Perspective 
Plane, 



Figure 1. 



Let A, B. C, D be the plan of the cube, S the sta- 
tion point, S. T the middle visual ray and B. L the 
base line, or perspective plane. 

1st. Continue the sides of the plan to the perspec- 
tive plane as shewn by the dotted lines, intersecting it 
in M. E. JVand 0. 

2nd. From the corners of the plan draw rays to the 
statian point, intersecting the perspective plane in a, d. 
b. c. 

3rd. Through S^ draw S, F parallel to A, D, and S. 
G parallel to D. C. Then F is the vanishing jjointjbr 
the sides A. D and B. Cj and G is the vanishing point 
for the sides A. B and D. C. 



Figure 2. 



4th. Transfer these intersections from B. X, fig. 1, 
to B. Ly fig. 2, and the vanishing points jP and G to 
the horizon, as shewn by the dotted lines. 

5th. From E and M^ draw lines to the vanishing 



128 PLATE XLIV. 

point G, and from JY and O, draw lines to the vanish- 
ing point F. Then the trapezium A. B. C. DJvrmed 
by the intersection of these lines j is the perspective view 
of the plan of the cube, 

6th. To DRAW THE ELEVATION. At M. E. JV and 
erect perpendiculars and make them equal to the 
side of the cube. 

7th. From the tops of these perpendiculars draw lines 
to the opposite vanishing points as shewn by the dotted 
lines, their intersection will form another trapezium 
parallel to the first, representing the top of the cube. 

8th. From ^, D and C, erect perpendiculars to com- 
plete the cube. 

Note. — It is not necessary to erect perpendiculars from all the 
points of intersection^ to draw the representation, but it is done 
here, to prove that the height of an object may be set on any per 
pendicular erected at the point where the plane, or line, of a con- 
tinuation of a line intersects the perspective plane ; one such line 
of elevation is generally sufficient. 

9th. To draw the figure with one line of heights, 
proceed as follows : from A, D and C, erect indefinite 
perpendiculars. 

10th. Make E, H equal to the side of the cube, and 
from H draw a line to G, cutting the perpendiculars 
from D and Cm K and L. 

11th. From iC, draw a line to P, cutting A. P in P; 
from L, draw a line to jP, and from P, draw a line to 
G, which completes the figure. 

Note. — ^The student should observe how the lines and hori- 
zontal planes become diminished as they approach toward the 
horizon, each successive line becoming shorter, and each plane 
narrower until at the height of the eye, the whole of the top 
would be represented by a straight line. I would here remark, 
that it would very materially aid the student in his knowledge 
of perspective, if he would always make it a rule to analyze the 
parts of every diagram he draws, observe the changes which 
take place in the forms of objects when placed in different posi- 




W^'MfjA^ 



I 



PLATE XLIV. 



129 



tions on the plan, and when they are placed above or below the 
horizon at different distances ; this would enable him at once to 
detect a false line, and would also enable him to sketch from 
nature with accuracy. Practice this always until it becomes a 
HABIT, and I can assure you it will be a source of much grati- 
fication. 



PLATE XLV. 



TO DRAW THE PERSPECTIVE VIEW OF A ONE 
STORY COTTAGE, SEEN ACCIDENTALLY. 



Figure 1 



Let A, B. C. D be the plan of the cottage, twenty 
feet by fourteen feet, drawn to the accompanying scale ; 
the shaded parts shew the thickness of the walls and 
position of the openings, the dotted lines outside 
parallel with the walls, give the projection of the roof, 
and the square E. F. G. fl, the plan of the chimney 
above the roof. 

Let P. L be the perspective plane and S the station 
point. 

1st. Continue the side B. D to intersect the per- 
spective plane in if, to find the position for aline of 
heights. 

2nd. From all the corners and jambs on the plan, 
draw rays toward the station point to intersect the per- 
spective plane. 

3rd. Through S draw a line parallel to the side of 
the cottage D. C, to intersect the perspective plane in 
L. This gives the vanishing point for the ends of the 
building and for all planes parallel to it, viz : the side 
of the chimney, and jambs of the door and windows. 



130 PLATE XLV. 

4th. Through S draw a line parallel with B. Dy to 
intersect the perspective plane, which it would do at 
some distance outside of the plate ; this intersection 
would be the vanishing point for the sides of the 
cottage, for the tops and bottoms of the windows, the 
ridge and eaves of the roof, and for the front of the 
chimney. 

Figure 2. 



Let us suppose the parallelogram P L. W. ^ to be a 
separate piece oj^ paper laid on the other, its top edge 
coinciding with the perspective plane of Jig » 1, and 
its bottom edge W. X to be the base of the picture, 
then proceed asfolloivs : 

1st. Draw the horizontal line R, T parallel to W, X 
and five feet above it. 

2nd. Draw H, Jl perpendicular to P. L for a line of 
heights. 

3rd. Draw a line from K to the vanishing point 
without the picture, w^hich we will call Z ; this will 
represent the line H. B of fig. 1, continued indefinitely. 

4th. From b and d draw perpendiculars to intersect 
the last line drawn, in o and e, which will determine the 
perspective length of the front of the house, 

5th. On jBl. H set off twelve feet the height of the 
walls, at 0, and from draw a line to the vanishing- 
point Z, intersecting d. e m m and 6. o in n, 

6th. From m and e draw vanishing lines to T, and 
a perpendicular from c intersecting them in F and s ; 
this will give the corner F, and determine the depth of 
the building, 

7th. Find the centre of the vanishing plane repre- 
senting the end, by drawing the diagonals m. F and e, s. 



Flate^ 45 . 
PLAN AND PERSPECTIVE VIEW. 




l^P': Mirpni' 



r.iTFum k:dotbs. 



PLATE XLV. 



131 



and through their intersection draw an indefinite per- 
pendicular u, v, which will give the position oj^ the gable. 

8th. To FIND THE HEIGHT OF THE GABLE, SCt off 

its proposed height, say 7' 0^', from to JV on the line 
of heights, from JY draw a ray to Z, intersecting e. d in 
Wy and from TTdraw a vanishing line to T intersecting 
u, V in v, then v is the peak of the gable. 

9th. Join m. v^ and prolong it to meet a perpen- 
dicular drawn through the vanishing point T, which it 
will do in F, then V is the vanishing point for the in- 
clined lines of the ends of the front half of the roof 
The ends of the backs of the gables will vanish in a 
point perpendicularly below V^ as much below the 
horizon as V is above it. 

10th. For the Roof. Through v draw v. y to Z 
without, to form the ridge of the roof, from/* let fall a 
perpendicular to intersect y. v in Wj through w draw a 
line to the vanishing point Fto form the edge of the 
roof. From d let fall a perpendicular to intersect V. 
Wy and from the point of intersection draw a line to Z 
to form the front edge of the roof, from a let fall a per- 
pendicular to define the corner x^ and from x draw a 
line to V intersecting w, y in y, which completes the 
front half of the roof; from w draw a line to the van- 
ishing point below the horizon, from c let fall a perpen- 
dicular to intersect it in g*, and through g draw a line 
to Z, which completes the roof 

11th. For the Chimney. Set off its height above 
the ridge at JkT, from M draw a line toward the vanish- 
ing point Z, intersecting o. b in C7, from U draw a line 
to the vanishing point T, which gives the height of the 
chimney, bring down perpendiculars from rays drawn 
from G. Fand £, fig. 1, and complete the chimney by 
vanishing lines drawn for the front toward Z and for 
the side toward T. 



132 PLATE XLV. 

12th. For the Door and Windows. Set off their 
heights at P. Q and draw lines toward Z, bring down 
perpendiculars from the rays as before, to intersect the 
lines drawn toward Z; these lines will determine the 
breadth of the openings. The breadth of the jambs are 
found by letting fall perpendiculars from the points of 
intersection, the top and bottom lines of the jambs are 
drawn toward T, 

Note I. — As the bottom of the front fence if continued, would 
intersect the base line at jK'the foot of the line of heights, and its 
top is in the horizon, it is therefore five feet high* 

Note 2. — ^Tlie whole of the lines in this diagram have been 
projected according to the rules, to explain to the learner the 
methods of doing so, and it will be necessary for him to do so 
until he is perfectly familiar with the subject. But if he will fol- 
low the rule laid down at the end of the description of the last 
plate, he will soon be enabled to complete his drawing by hand, 
after projecting the principal lines, but it should not be attempted 
too early, as it will beget a careless method of drawing, and pre- 
vent him from acquiring a correct judgment of proportions. 



PLATE XLVI 



THE FRONTISPIECE 

Is a perspective view of a street sixty feet wide, as 
viewed by a person standing in the middle of the 
street at a distance of 134 feet from the perspective 
plane, and at an elevation of 20 feet from the ground to 
the height of the eye. The horizon is placed high for 
the purpose of shewing the roofs of the two story 
dwellings. 



PLATE XLVI. 133 

The dimensions of the different parts are as follows : 
1st* — ^Distances across the Picture. 
Centre street between the houses 60 ft. wide. 

Side walks, each 10 " 

Middle space between the lines of railway 4 6 " 
Width between the rails 4 9 " 

Depth of three story warehouse 40 feet. 

Depth of yard in the rear of warehouse 20 " 
Depth of two story dwelling on the right 30 



iC 



Distances from the Spectator, in the Line of 
THE Middle Visual Ray. 

From spectator to plane of the picture 134 feet. 

From plane of picture to the corner of buildings 50 " 
Front of each house 20 " 

Front of block of 7 houses 20 feet each 140 " 



60 



Breadth of street running across between } 
the blocks ^ 

Depth of second block same as the first 140 ^' 

Depth of houses on the left of the picture, ) .^ 
behind the three story warehouses ) 



c< 



a 



To Draw the Picture. 

1st. Let C be the centre of the perspective plane, 
H. L the horizon, S the point of sight. 

2nd. From C on the line P. P, lay off the breadth 
of the street thirty feet on each side, at and 60, 
making sixty feet, and from those points draw rays to 
the point of sight ; these give the lines of the fronts of 
the houses. 

3rd. From lay off a point 50 feet on P. P, and 

draw a diagonal from that point to the point of distance 

without the picture ; the intersection of that diagonal 

with the ray from 0, determines the corner of the 
_ 



134 PLATE XLVI. 

building; from the point of intersection erect a perpen- 
dicular to jB. 

4th. From 50, lay off spaces of 20 feet each at 70, 
90 and so on, and from the points so laid off draw 
diagonals to determine by their intersection with the 
ray from 0, the depth of each house. 

5th. After the depth on 0. S is found for three 
houses, the depths of the others may be found by draw- 
ing diagonals to the opposite point of distance to inter- 
sect the ray 60 aS', as shewn by the dotted lines. 

Note. — As a diagonal drawn to the point of distance forms an 
angle of 45^ with the plane of the picture, it follows that a diago- 
nal drawn from a ray to another parallel ray, will intercept on 
that ray a space equal to the distance between them. Therefore 
as the street in the diagram is 60 feet wide and the front of each 
house is 20 feet, it follows that a diagonal drawn from one side 
of the street to the other will intercept a space equal to the fronts 
of three houses, as shewn in the drawing. 

6th. Lay off the dimensions on the perspective plane, 
of the depth of the houses, and the position of the 
openings on the side of the warehouse, and draw rays 
to the point of sight as shewn by the dotted lines. 

7th. At erect a perpendicular to D for a line of 
heights ; on this line all the heights must be laid off to 
the same scale as the measures on the perspective 
plane, and from the points so marked draw rays to the 
point of sight to intersect the corner of the building at 
J9. For example, the height of the gable of the ware- 
house is marked at Ji^ from .^ draw a ray toward the 
point of sight intersecting the corner perpendicular at 
B; then from jB, draw a horizontal line to the peak of 
the gable ; the dotted lines shew the position of the 
other heights. 

8th. To find the position of the peaks of the gables 
on the houses in the rear of the warehouse, draw rays 



I 



PLATE XLVI. 135 

from the top and bottom corner of the front wall to the 
point of sight, draw the diagonals as shewn by the 
dotted lines, and from their intersection erect a perpen- 
dicular, which gives the position of the peak, the inter- 
section of diagonals in this manner will always deter- 
mine the perspective centre of a vanishing plane. The 
height may be laid off on 0. D at D, and a ray drawn 
to the point of sight intersecting the corner perpen- 
dicular at (7, then a parallel be drawn from C to inter- 
sect a perpendicular from the front corner of the build- 
ing at Ej and from that intersection draw a ray to the 
point of sight. The intersection of this ray, with the 
indefinite perpendicular erected from the intersection of 
the diagonals, will determine the perspective height of 
the peak. 

9th. The front edges of the gables will vanish in a 
point perpendicularly above the point of sight, and the 
back edges in a point perpendicularly below it and 
equidistant. 

10th. As all the planes shewn in this picture except 
those parallel with the plane of the picture are parallel 
to the middle visual ray, all horizontal lines on any of 
them must vanish in the point of sight, and inclined 
lines in a perpendicular above or below it, as shewn by 
the gables. 



136 SHADOWS. 



SHADOWS. 



1st. The quantity of light reflected from the surface 
of an object, enables us to judge of its distance, and 
also of its form and position. 

2nd. On referring to paragraph 9, page 98, it will 
be found that light is generally considered in three 
degrees, viz : lights shade and shadow ; the parts 
exposed to the direct rays being in light, the parts in- 
clined from the direct rays are said to be in shade, and 
objects are said to be in shadow, when the direct rays _ 
of light are intercepted by some opaque substance being 
interposed between the source of light and the object. 

3rd. The form of the shadow depends on the form 
and position of the object from which it is cast, modi- 
fied by the form and position of the surface oh which 
it is projected. For example, if the shadow of a cone 
be projected by rays perpendicular to its axis, on a 
plane parallel to its axis, the boundaries of the shadow 
will be a triangle; if the cone be turned so that its 
axis be parallel with the ray, its shadow will be a 
circle ; if the cone be retained in its position, and 
the plane on which it is projected be inclined in either 
direction, the shadow will be an ellipsis, the greater 
the obliquity of the plane of projection, the more elon- 
gated will be the transverse axis of the ellipsis. 

4th. Shadows of the same form may be cast by 
DIFFERENT FIGURES : for example, a sphere and a flat 
circular disk would each project a circle on a plane 
perpendicular to the rays of light, so also would a cone 



SHADOWS. 137 



and a cylinder with their axes parallel to the rays. 
The sphere would cast the same shadow if turned in 
any direction, but the flat disk if placed edgeways to 
the rays, would project a straight line, whose length 
would be equal to the diameter of the disk and its 
breadth equal to the thickness ; the shadow of the cone 
if placed sideways to the rays would be a triangle, and 
of the cylinder would be a parallelogram. 

5th. Shadows of regular figures if projected on a 
plane, retain in some degree the outline of the object 
casting them, more or less distorted, according to the 
position of the plane ; but if cast upon a broken or 
rough surface the shadow will be irregular. 

6th. Shadows projected from angular objects are 
generally strongly defined, and the shading of such 
objects is strongly contrasted; thus if you refer to the 
cottage on Plate 45, you will perceive that the vertical 
walls of the front and chimney are in light, fully ex- 
posed to the direct rays of the sun, while the end of 
the cottage and side of the chimney are in shade, being 
turned away from the direct rays, the plane of the roof 
is not so bright as the vertical walls, because, although 
it is exposed to the direct rays of light it reflects them 
at a different angle, the shadow of the projecting eaves 
of the roof on the vertical wall forms a dark unbroken 
line, the edge of the roof being straight and the sur- 
face of the front a smooth plane, the under side of the 
projecting end of the roof is lighter than the vertical 
wall because it is so situated as to receive a larger pro- 
portion of reflected light. 

7th. Shadows projected from circular objects are 
also generally well defined, but the shadings instead of 
being marked by broad bold lines as they are in rect- 
angular figures, gradually increase from bright light 

to the darkest shade and again recede as the opposite 
_ 



138 SHADOWS. 



side is modified by the reflections from surrounding ob- 
jects, so gradually does the change take place that it 
is difficult to define the exact spot where the shade 
commences, the lights and shades appear to melt into 
each other, and by its beautifully swelling contour ena- 
bles us at a glance to define the shape of the object. 

8th. Double Shadows. — Objects in the interior of 
buildings frequently cast two or more shadows in op- 
posite directions, as they receive the light from oppo- 
site sides of the building ; this effect is also often pro- 
duced in the open air by the reflected light thrown from 
some bright surface, in this case however, the shadow 
from the direct rays is always the strongest ; in a room 
at night lit by artificial means, each light projects a 
separate shadow, the strength of each depending on 
the intensity of the light from which it is cast, and its 
distance from the object ; the student may derive much 
information from observing the shading and shadows 
of objects from artificial light, as he can vary the angle, 
object and plane of projection at pleasure. 

9th. The extent of a shadow depends on the angle 
of the rays of light. If we have a given object and 
plane on which it is projected, its shadow under a 
clear sky will vary every hour of the day, the sun's 
rays striking objects in a more slanting position in the 
morning and evening than at noon, projects much long- 
er shadows. But in mechanical or architectural draw- 
ings made in elevation, plan or section, the shadows 
should always be projected at an angle of 45^, that is 
to saj% the depth of the shadow^ should always be equal 
to the breadth of the projection or indentation; if this 
rule is strictly followed, it will enable the workman to 
apply his dividers and scale, and ascertain his projec- 
tions correctly from a single drawing. 



PLATE XLVII. 139 

Note. — ^The best method for drawing lines at this angle^ is to 
use with the T square, a right angled triangle with equal sides, 
the hypothenuse will be at an angle of 45^ with the sides ; with 
the hypothenuse placed against the edge of the square, lines may 
be drawn at the required angle on either side. 



PLATE XLVII. 

PRACTICAL EXAMPLES FOR THE PROJECTION 
OF SHADOWS. 

Figure 1 



Is a square shelf supported by two square bearers 
projecting from a wall. The surface of the paper to 
represent the wall in all the following diagrams, 

1st. Let A. J5. a D be the plan of the shelf; A. B 
its projection from the line of the wall W, X; B. D the 
length of the front of the shelf, and E and F the plans 
of the rectangular bearers. 

2nd. Let G. H be the elevation of the shelf shewing 
its edge, and J and K the ends of the bearers. 

3rd. From all the projecting corners on the plan, 
draw lilies at an angle of 45^ to intersect the line of 
the wall W. Z, and from these intersections erect in- 
definite perpendiculars. 

4th. From all the projecting corners on the eleva- 
tion, draw lines at an angle of 45° to intersect the per- 
pendiculars from corresponding points in the plan; the 
points and lines of intersection define the outline of the 
shadow as shewn in the diagram. 



140 PLATE XLVII, 



Figure 2 



Is a square Shelf against a wall supported by two square 

Uprights. 

L. M. JY. is the plan of the shelf, P and Q the 
plans of the uprights, R. S the front edge of the shelf, 
Tand Fthe fronts of the uprights. 

1st. From the angles on the plan draw lines at an 
angle of 45^ to intersect W, JST, and from the intersec- 
tions erect perpendiculars. 

2nd. From R and S, draw lines at an angle of 45° 
to intersect the corresponding lines from the plan. 

Figure 3 



Is a Frame with a semicircular head^ nailed against a 
wall^ the Frame containing a sunk Panel of the same 
form. 

1st. Let A, B, C. D be the section of the frame and 
panel across the middle, and F on the elevation of the 
panel, the centre from which the head of the panel and 
of the frame is described. 

2nd. From JS, draw a line to intersect the face of the 
panel, and from D to intersect W. Xy and erect the per- 
pendiculars as shewn by the dotted lines. 

3rd. From JVand JV,' draw lines to define the bot- 
tom shadow, and at L draw a line at the same angle 
to touch the curve. 

4th. At the same angle draw F. G, make F. H equal 
to the depth of the panel, and F. G equal to the thick- 
ness of the frame. 

5th. From H with the radius JP. il, describe the 
shadow on the panel, and from G with the radius F. S^ 
describe the shadow of the frame. 



Fla;t£. 47. 



SHADOWS 



Fig. ? 



W A 





Fig. 3.. 



W 




A! 






c 


Bl2i 


\ 




•^T) 



Tig .4-. 



W- 




Fig.c 




.-,■// A:'-^ iz: 



PLATE XLVII. 141 

Note. — ^The tangent drawn at L and the curve of the shadow 
touch the edge of the frame in the same spot, but if the propor- 
tions were different they would not do soj therefore it is always 
better to draw the tangent. 



Figure 4 



Is a Circular Stud representing an enlarged view of one 
of the JVdil Heads used in the last diagram, oj^ which 
N. O. P 15 a section through the middle y and W. X 
the face of the frame. 

1st. Draw tangents at an angle of 45° on each side 
of the curve. 

2nd. Through L the centre, draw L. Jlf, and make 
L. M equal to the thickness of the stud. 

3rd. From Jlf, with the same radius as used in de- 
scribing the stud, describe the circular boundary of the 
shadow to meet the two tangents, which completes the 
outline of the shadow. 



Figure 5 



Is a Square Pillar standing at a short distance in front 
of the wall W. X. 

1st. Let A. B, C jD be the plan of the pillar, and 
W. X the front of the wall, from A. C. D draw lines to 
W. X, and from their intersections erect perpendiculars. 

2nd. Let E. F. G. H be the elevation of the pillar, 
from F draw F. K. L to intersect the perpendiculars 
from C and D. 

3rd. Through if, draw a horizontal line, which com- 
pletes the outline. The dotted lines shew the position 
of the shadow on the wall behind the pillar. 



L 



142 PLATE XLVIII. 



PLATE XLVIII. 
SHADOWS — CONTINUED 

Figure 1 



Is the Elevation and Fig. 2 the Plan of a Flight of 
Steps with rectangular Blockings at the endsy the 
edge of the top step even with the face of the wall. 

1st. From Jl. B. C and D, draw lines at an angle 
of 45^. 

2nd. From F where the ray from C intersects the 
edge of the front step, draw a perpendicular to JV, 
which defines the shadow on the first riser. 

3rd. From Q where the ray from C intersects the 
edge of the second step, draw a perpendicular to JIf, 
which defines the shadow on the second riser. 

4th. From K where the ray from A intersects the 
top of the third step, draw a perpendicular to 0, which 
defines the shadow on the top of that step. 

5th. From L where the ray from A intersects the top 
of the second step, draw a perpendicular to H inter- 
secting the ray drawn from Cin fi, which defines the 
shadow on the top of the second step. 

6th. From P where the ray from B intersects the 
ground line, draw a perpendicular to intersect the ray 
drawn from D in E ; this defines the shape of the 
shadow on the ground. 



^ 



FUite 48 



SHylDOtVS . 




B DF 



M F,-y 




yr- Wmfij',. 



I 



TLbiiaji kSoru 



PLATE XLVIII. 143 



Figure 3, 



To draw the Shadow of a Cylinder upon a Vertical 

Plane, 

Rule. — Find the position of the shadow at any num- 
ber of points. 

1st. From A where the tangental ray (at an angle 
of 45°) touches the plan, draw the ray to W, Xy and 
from the intersection erect a perpendicular. 

2nd. From A erect a perpendicular to B, and from 
B draw a ray at 45*^ with A. B to intersect the perpen- 
dicular from Ji in L. This defines the straight part of 
the shadow. 

3rd. From any number of points in the plan E. fZ, 
draw rays to intersect the wall line W, X^ and from 
these points of intersection erect perpendiculars. 

4th. From the same points in the plan erect perpen- 
diculars to the top of the cylinder, and from the ends 
of these perpendiculars draw rays at 45° to meet the 
perpendiculars on the wall line ; the intersections give 
points in the curve. 

Note 1. — The outlines of shadows should be marked by faint 
lines, and the shadow put on by several successive coats of India 
ink. The student should practice at first with very thin color, 
always keep the camel hair pencil full, and never allow the edges 
to dry until the whole shadow is covered. The same rule will 
apply in shading circular objects ; first wash in all the shaded 
parts with a light tint, and deepen each part by successive layers, 
always taking care to cover with a tint all the parts of the object 
that require that tint ; by this means you will avoid harsh out- 
lines and transitions, and give your drawing a soft agreeable 
appearance. 

Note 2. — ^The lightest part of a circular object is where a tan- 
gent to the curve is perpendicular to the ray as at P. The darkest 
part is at the point where the ray is tangental to the curve as at 
A, because the surface beyond that point receives more or less 
reflected light from surrounding objects. 



INDEX. 






PACE. 


Abscissa, . . 


. 60 


Jlbsorption of light, .... 


97 


Accidental points, . 


. 107 


Aerial perspective, . . 


102 


Altitude of a triangle. 


. 14 


Angles described, . ... 


12 


Angle of incidence. 


. 97 


" Visual ..... 


100 


*' How to draw angles of 45^, 


. 139 


Apex of a pyramid, .... 


54 


** of a cone, .... 


. 61 


Application of the rule of 3, 4 and 5, . 


24 


Apparent size of an object. 


. 108 


Arc of a circle, . , . . , 


17 


Arcades in perspective. 


. 119 


Arches — Composition of . . . 


72 


" Construction of . 


. 72 


** Definitions of .... 


73 


Arch — ^Thrust of an 


. 73 


" Amount of the thrust of an arch, (note) 


74 


" Straight arch or plat band, . 


. 74 


*' Rampant .... 


74 


'' Simple and complex arches. 


. 74 


^' Names of arches, . . . 


74 


Arches in perspective. 


. lis to 123 


Arithmetical perspective. 


99 


w^aji* of a pyramid. 


. 54 


" of a sphere, .... 


57 


" of a cylinder, .... 


. 58 


** Major and minor axes, . 


. 59—60 


** of a cone, . . . 


. 61 


"' Difference between the length of the axes in 


the sec- 


tions of the cone and cylinder, . 


63 


*' of the parabola. 


. 69 


Axioms in perspective, . 


108 


Back of an arch or Extrados, 


. 73 


Band, listel or fillet, . . 


81 



13 



146 INDEX. 




PAGE. 


Base of a triangle, ..... 


. 14 


" "of a pyramid, .... 


54 


*' of a cone. 


. 61 


*' of a Doric column. 


82 


'' line or ground line, . 


. 103 


Bead described, . . , . 


82 


Bed of an arch, .... 


. 74 


Bisect — To bisect a right line, . 


20 


" To bisect an angle. 


. 26 


Bird's eye view, .... 


106 


Cavetto, a Roman moulding. 


. 83 


Centre of a circle, .... 


17 


'• of a sphere. 


. 56 


*^ of a vanishing plane, . . . 


135 


CJwrds defined, .... 


. 18 


" Scale of chords constructed. 


36 


" Application of the scale of chords, . 


. 37 


Circle described. 


17 


^' To find the centre of a circle. 


. 31 


*' To draw a circle through three given points 


>, . 32 


*^ To find the centre for describing a flat segm 


ent, . 32 


" To find a right line equal to a semicircle, . 


33 


" " '^ '' equal to an arc of a circ 


le, . 33 


'' Workmen's method of doing the same. 


34 


" Great circle of a sphere. 


. 57 


" Lesser circle of a sphere. 


57 


*^ Circumferences of circles directly as the diai 


neters, . 88 


" in perspective, .... 


114 


" Application of the circle in perspective. 


. 123 


Circular plan and elevation. 


81 


" objects — Shading of 


. 137 


Circumference of a circle. 


17 


'^ " directly as its diameter. 


. 88 


Circular domes — To draw the covering of 


70 


Color — How to color shadows, &c. 


. 143 


Conjugate axis or diameter, : 


. 59—60 


" of a diameter of the ellipsis. 


. 60 


Contents of a triangle, .... 


15 


" of a cube, .... 


51—52 


" of the surface of a cube. 


53 


Complement of an arc or angle. 


. 19 



INDEX. 


147 




VAQE. 


Complex and simple arches, . . . 


74 


Cone, right, oblique and scalene, . 


. 56 


" To draw the covering of a cone. 


56 


'^ Sections of the cone. 


. 62 


Co-sine . . . . 


19 


Co-tangent, . . . » , 


. 19 


Co-secant, ..... 


20 


Construct— 'To construct a triangle. 


. 25 


" " an angle equal to a given angle, 25 


'' " an equilateral triangle on a given line, 27 


" "a square on a given line, . 28 


'^ " a pentagon on a given line, . 30 


" " a heptagon on a given line, . 31 


" '^ any polygon on a given 


line, . 31 


" *^ a scale of chords. 


36 


*^ ^^ the protractor. 


. 42 


Construction of arches. 


72 


Contrary flexure — Curve of 


. 12 


Arch of 


76 


Cottage in perspective 


. 129 


Covering of the cube, . . . 


53 


" " parallelopipedon, . 


. 64 


" " triangular prism. 


54 


" " square pyramid, . 


. 54 


" " hexagonal pyramid, . 


55 


" " cylinder, . 


. 55 


'' cone. 


56 


" " sphere. 


. 57 


" " regular polyhedrons. 


57 


*' ^' circular domes. 


.70 


Crown of an arch, . . . . 


73 


" moulding, . . . 


. 83 


Cube or hexahedron. 


52 


Cwfeic measure, .... 


. 51 


Cvhe — ^To draw the Isometrical 


89 


^^ in perspective. 


. i2r 


CycZotd described, . . . 


49 


Cycloidal arches, . . 


. 50 


Cylinder, ..... 


55 


" To draw the covering of a 


. 56 


" Sections of the . . . 


59 



148 INDEX. 


j' 




PAGE. 


Cylinder — Right and oblique 


. 68 


j 


" in perspective. 


124 




" Shadow of a . 


. 143 




Cyma or cyma recta — Roman, . 


83 




" '^ Grecian, . . 


. 85 




'^ reversa, talon or ogee — Roman, 


84 




" " '' " Grecian, \ 


. 86 




Degree defined. 


18 




Depressed arch. 


. 79 




Diamond defined, . . • . 


15 




Diagonal defined, ..... 


. 16 




Diagonal lines in perspective, . . 


106 




Diameter of a circle, . . 


. 17 




^' of a sphere, .... 


57 




" of an ellipsis. 


59—60 




" of the parabola. 


69 




De^mfions of lines, .... 


. 11 




'^ of angles, .... 


12 




" of superficies, . . . . 


, 13 




" of the circle. 


17 




" of solids, . . . 


. 61 




" of the cylinder. 


58 




" of the cone. 


. 61 




^* of the parabola. 


69 


\ 


" of arches, . . 


. 73 




" in perspective. 


103 




Directrix of the parabola, .... 


. 69 




Distance — Points of . 


104 




Half . . 


. 110 




" The quantity of reflected light enables us 


to 




judge of .... 


136 




Dodecahedron, ..... 


. 58 




Domes — Covering of hemispherical 


70 




Double shadows, ..... 


. 138 




Echinus, or Grecian ovolo. 


84 




Ellipsis— Fsihe ... 


48—49 




'' the section of a cylinder. 


59 




^^ To describe an ellipsis with a string 


. 59 




" the section of a cone. 


62 




" To describe an ellipsis from the cone 


. 62 




" To describe an ellipsis by intersections 


64 





INDEX. 




149 






PAGE. 


Ellipsis — To describe an ellipsis with a trammel . 




65 


Elliptic Arch — ^To draw the joints of an 




77 


Epicycloid described, .... 




50 


Equilateral triangle, .... 




14 


" arch, (Gothic) 




78 


Extrados or back of an arch. 




73 


Fillet, band or listel, . . , 




81 


Focus — Foci of an ellipsis. 




59 


" of a parabola, .... 




69 


Foreshortening, ..... 




100 


^' The degree of foreshortening depends on the 




angle at which objects are viewed. 


101- 


-108 


Form of shadows, .... 




136 


Frustrum of a pyramid, .... 




54 


'^ of a cone, .... 




62 


Globe or sphere, ..... 




56 


Gothic arches described. 




78 


Grades, ...... 




19 


Grecian mouldings, .... 




84 


Ground line or base line, . . 




103 


Habit of observation, . 




128 


Half distance, ..... 




110 


Height, rise or versed sine of an arch, . 




73 


Hemisphere, ..... 




57 


Hexahedron or cube, .... 




52 


Hexagonal pavement in perspective. 




113 


Horizontal or level line, . . 




12 


" covering of domes. 




71 


Horizon in perspective, . . . . 




103 


Horseshoe arch. 




76 


'' pointed arch, .... 




79 


Hyperbola the section of a cone. 




62 


"' to describe the hyperbola from the cone. 


. 63 


66 


Hypothenu^e, . . . 


14 


—23 


" Square of the . 




23 


Icosahedron, . . . 




58 


Inclined lines in Isometrical drawing require a ( 


lifferent 




scale. 




96 


Incidence — ^The angle of incidence equal to the 


angle of 




reflection. 


. 


97 


Inclined lines — Vanishing point of . 




108 



13* 



150 INDEX. 






PAGE. 


Inclined planes — Vanishing point of . 


107 


Inscribe — To inscribe a circle in a triangle. 


. 27 


*' " an octagon in a square, . 


28 


*' " an equilateral triangle in £ 


I circle, . 29 


" '' a square in a circle. 


30 


" *' a hexagon in a circle. 


. 29 


'^ " an octagon in a circle. 


30 


" " a dodecagon in a circle. 


. 29 


Intrados or soffit of an arch, . 


73 


Isometrical drawing, 


. 89 


" cube, . . 


89 


" circle, .... 


. 94 


Isometrical circle — To divide the 


95 


Isosceles triangle, .... 


. 14 


JointF of an arch defined, 


74 


*^ To draw the joints of arches. 


75 to 80 


Keystone of an arch. 


73 


Lancet Arch — To describe the 


. 78 


Light — Objects to be seen must reflect light. 


96 


*' becomes weaker in a duphcate ratio, &c 


;., . . 96 


" Three degrees of 


. 98—136 


Lines — Description of . . . 


. 11 


Line — To divide a right 


35 


" To find the length of a curved 


. 33 


'' Workmen's method of doing so, 


34 


Line of centres (of wheels,) 


. 87 


'' Pitch line defined, 


88 


" To draw the pitch line of a pinion to c 


ontain a 


definite number of teeth. 


. 88 


" Ground or base line. 


103 


'^ Vanishing point of a line, . 


. 108 


'^ of elevation in perspective. 


115 


Linear perspective defined, . 


. 102 


Listel, band or fillet, . 


81 


Lozenge defined, .... 


. 15 


Major and minor axes or diameters. 


. 59--60 


Measures — Cubic, .... 


. 51 


" Lineal and superficial. 


51 


" of the surface of a cube, 


. 53 


Middle ray or central visual ray. 


105 


Minutes, ..... 


. 18 



INDEX. 


151 




PAOE. 


Mitre — ^To find the cut of a . . . 


. 26 


Moresco or Saracenic arch. 


76 


Mouldings — Roman , . . . 


. 81 


" Grecian . • . . 


84 


Obelisk defined, .... 


. 55 


Oblique pyramid, .... 
'* cone, ..... 


54 
56—61 


" cylinder, . . v . . 
Oblong defined. 
Octagonal plan and elevation, . 


59 
. 15 

80 


Octahedron, ..... 


. 57 


Ogee Arch or arch of contrary flexure, . 
Ogee or cyma reversa — Roman 


76 
. 83 


Grecian 


86 


Optical illusion. 

Ordinate of an ellipsis. 

Ovals composed of arcs of circles, , 


. 91 

60 

47—48 


Ovolo — Roman .... 


82 


" or Echinus — Grecian 


. 84 


Parallelogram defined, . 


15 


Parallel lines, .... * 


. 12 


** ruler, ..... 


34 


" Application of the parallel ruler, . 
Parallelopipedon, .... 
1 Parabola— To find points in the curve of the 


. 35 

54 

. 45 


" the section of a cone. 


62 


i " To describe a parabola from the cone, . 


. 63 


** To describe a parabola by tangents, &c. . 67 
" To describe a parabola by continued motion, . 68 
^' applied to Gothic arches, ... 69 


" Detinitions of the parahola. 


. 69 


Parameter defined, .... 


69 


Pentagon reduced to a triangle. 


. 36 


" To construct a pentagon on a given line. 

Perimeter the boundary of polygons. 

Periphery the boundary of a circle. 

Perpendicular lines defined, 

" To bisect a line by a perpendicular, 
^' To erect a perpendicular, 
" To let fall a perpendicular. 

Perspective — Es§ay on perspective, 


30 
. 14 

17 
. 12 

20 
20—21—23 

22 
. 96 



152 INDEX. 

* „ ■■- ■ 




PAOB. 


Perspective— h'mesir and aeriel perspective. 


102 


" plane, or plane of the picture, 


103 


" ** must be perpendicular to the middle 


visual ray, 


105 


" plan of a square, , , .109 


—110 


" '* of a room with pilasters, 


HI 


** " To shorten the depth of a perspective plan, 112 { 


" Tesselated pavements in perspective, , 


112 1 


" Double square in perspective, . . 


114 1 


" Circle in perspective, 


114 


*^ Line of elevation, .... 


115 


** Pillars with projecting caps in perspective. 


116 


'* Pyramids in perspective, . . 


117 


" Arches seen in front. 


118 


** /* on a vanishing plane, 


121 


*' Application of the circle, . 


123 


** To find the perspective plane, &,c. 


125 


"^ view of a cube seen accidentally, . 


127 


*' view of a cottage seen accidentally, . 


129 


" view of a street, . 


132 


Pillars m perspective, . . 


116 


Pitch of a wheel, ..... 


'88 


" circle of a wheel, ..... 


87 


*' line of a wheel, ..... 


88 


Plane superficies, ...... 


13 


Planes — Vanishing ..... 


106 


" Parallel planes vanish to a common point. 


. 108 


'' parallel to the plane of the picture. 


108 


*' To find the perspective plane. 


125 


Plat hand or straight arch, .... 


74 


Platonic figures, ...... 


58 


Poinf of intersection. 


19 


" of contact, ..... 


19 


" Secant point, . 


19 


'' of sight, . . 


104 


*' of view or station point, .... 


104 


" Vanishing points, .... 


. 105 


'' Principal vanishing point, . 


105 


'' of distance, . 


. 104 


Pointed arches in perspective, .... 


119 


Poles of the sphere, . . . . . 


. 57 



INDEX. 


153 




PAGE. 


Proportional diameter of a wheel. 


87 


'' circle or pitch line. 


. 88 


Polygons described, .... 


14 


" Table of polygons. 


. 27 


" Regular and irregular polygons. 


28 


Polyhedrons, .... 


52—57 


Projecting caps in perspective, . 


117 


Protractor — Construction of the protractor. 


. 42 


" Application of the protractor. 


42 


Prisms, ..... 


. 54 


Pyramid, ..... 


54 


" in perspective, . . 


. 117 


Quadrant of a circle, .... 


18 


Quadrangle defined. 


. 15 


Quadrilateral defined, .... 


15 


Radius — Radii, .... 


.17 


Rampant arch, . . 


74 


Rays of light reflected in straight lines. 


. 98 


** converged in the crystalline lens. 


99 


Rectangle defined, .... 


. 15 


Reduce — To reduce a trapezium to a triangle, . 


35 


" To reduce a pentagon to a triangle. 


. 36 


Reflection of light, .... 


96 


'^ The angle of reflection equal to the 


angle of 


incidence. 


. 97 


Reflected light enables us to see objects not illuminated by | 


direct rays, .... 


98 


Regular triangles, .... 


. 14 


" polyhedrons, . . 


. 52—57 


Refraction of light. 


. 96 


Retina of the eye, . . . 


99 


Rhomb — Rhombus, . . 


. 15 


Rhomboid, . . . . . 


15 


Right angled triangle, . . 


. 14 


Right line defined. 


11 


" pyramid, . . . 


. 54 


" cyhnder, ..... 


55 


^' cone, .... 


56—61 


Rise or versed sine of an arch, . 


73 


Roman mouldings. 


. 81 


Rule of 3, 4 and 5, . 


. 23—24 



154 INDEX. 




' 






PAGE. 


Saracenic or Moresco arch. 


, 


76 


Scale of chords, .... 


, 


36 


Scales of equal parts, . . . . 


. 


38 


'' Simple and diagonal scales. 


39—40 


—41 


*^ Proportional scale in perspective. 


. 


116 


Scalene triangle, .... 


. 


14 


Scheme or segment arch. 


, 


75 


Scotia described — Roman, . 


, , 


82 


'' " Grecian, 


, 


87 


Secant — Secant point, or point of intersection. 


. 


19 


Seconds, ..... 


, 


18 


Sector of a circle, . 


. 


18 


Sections of the cyHnder, 


, 


58 


" of the cone. 


61 to 67 1 


" of the eye, . . . . 


. 


99 


Serpentine line. 


. 


12 


Segment of a circle, . . . 


.. 


17 


'' To find the centre for describing a segment. 


32 


" To find a right line equal to a segment of 


a circle 


33 


" To describe a segment with a triangle. 




44 


" To describe a segment by intersections. 


, , 


46 


" of a sphere, .... 




57 


" or scheme arch, . . 


, , 


75 


Semicircle, . . . , 




17 


Semicircular arch, .... 


, 


76 


" '' in perspective. 




119 


Shade and shadow. 


98- 


-136 


Shadow always darker than the object. 


. 98—108 


Shadows — Essay on shadows. 


, , 


136 


SJmding of circular objects. 




137 


Shadow — Lightest and darkest parts of a . 


. 


143 


Sight — Method of sight. 


. 96-99 


'' Point of sight. 


, , 


104 


Simple and complex arches. 




74 


Sine, ..... 


, , 


19 


<Sfcei«-Z?ac/c of an,arch, .... 




75 


Soffit or intrados of an arch. 


, , 


73 


Span of an arch, .... 




73 


Sphere — Definitions of the sphere. 


. . 


56 


" To draw the covering of a sphere. 




57 


Springing line of an arch, . 


■ 


73 



INDEX. 


155 




FACE. 


Square, ...... 


15 


Square corner in a semicircle. 


. 21 


^' '' by scale of equal parts, . 


23 


*' of a number, . . . . 


. 23 


" of the hypothenuse. 


23 


Station point or point of view, 


. 104 


Straight or right line, .... 


11 


" arch or plat band, . , . , 


. 74 


Street in perspective, .... 


132 


Subtense or chord, . 


. 18 


iSwmmif of an angle, 


12 


" of a pyramid, .... 


. 54 


•' of a cone, . . . 


. 56--61 


Superficies or surface, . . . . 


. 13 


Supplement of an angle or arc, . 


19 


Table of the names of polygons, . . . 


. 27 


" " the angles of polygons. 


. Plate 10 


Talon or Ogee — Roman, .... 


. 83 


Grecian, 


86 


Tangent defined, ..... 


. 19 


Teeth of wheels — To draw the teeth of wheels. 


87 


Pitch of the '' . . 


. 88 


** Depth of the " . 


88 


Tesselated pavements in perspective. 


. 112 


Tetragon defined, .... 


15 


Tetrahedron one of the regular solids. 


. 57 


Torus described, .... 


82 


Trapezium defined, .... 


. 15 


" reduced to a triangle. 


35 


Trapezoid defined, ..... 


. 15 


Traiwverse axis or diameter. 


. 59—60 


Trigons or triangles, . . 


. 14 


Trisect — To trisect a right angle. 


26 


Truncated pyramid, . . . 


. 54 


*' cone, .... 


62 


Tudor or four centered arch. 


. 79 


Vanishing points, . . . 


105 


" Principal vanishing point, 


. 105 


" planes, .... 


106 


Versed sine of an arc, .... 


. 19 


*' '^ or rise of an arch, . 


73 



156 INDEX. 










PAGE. 


Verttx of a triangle, . . 




. 14 


*' of a pyramid. 




54 


'^ of a cone, . 




. 60 


" of a diameter of the ellipsis. 




60 


'* Principal vertex of a parabola. 




. 69 


" of a diameter of the parabola, . 




69 


Vertical or plumb line. 




. 12 


*' coverings of domes. 




70 


Vimal angle. 




. 100 


" rays, .... 




105 


Fbi<ssotr» of an arch. 


. , 


. 73 


Wlieel and pinion~To proportion the teeth of a 


88 


Wheel viewed in perspective. 


. 


. 101 




MINIFIE'S TEXT BOOK 

MECHANICAL DRAWING, 

FOR SELF-IINSTRUCTION; 

CONTAINING, 

1st. A series of progressive- practical problems in Geometry, 
with full explanations, couched in plain and simple terms; show- 
ing also the construction of the Parallel. Ruler, Plane 
Scales and Protractor. 

2nd. Examples for drawing Plans, Sections and Elevations 
of Buildings and Machinery, the mode of drawing elevations from 
Circular and Polygonal plans, and the Drawing of Roman 
and Grecian Mouldings. 

3rd. An introduction to Isometrical Drawing. 

4th. A treatise on Linear Perspective, with numerous ex- 
amples and full explanations, rendering the study of the art easy 
and agreeable. 

5th. Examples for the projection of shadows. 

the whole illustrated with 

FIFTY-SIX STEEL PLATES, 

CONTAINING 

Over Two Hundred Diagrams. 



BY 

W^M. MINIFIB, Architect, 

AND TEACHER OF 

DRAWING IN THE CENTRAL HIGH SCHOOL 
OF BALTIMORE. 

Price $3. 00. 

PUBLISHED BY WM. MINIFIE & CO. 

No. 114 BALTIMORE STREET, 

BALTIMORE. 

1849. 

1 



OPINIONS OF THE PRESS. 



From the Baltimore American, 

**We have examined with pleasure a very useful work, entitled 
A Text Book of Geometrical Drawing, just published by our 
townsman, Mr. Wm. Minifie, Architect, and Teacher of Drawing 
in the Central High School of Baltimore. The work is designed 
for the use of Mechanics and Schools, and has also been prepared 
for those who desire to instruct themselves. For this purpose the 
author has taken care to employ the most simple terms in his defi- 
nitions as well as his problems, and has illustrated his lessons for 
drawing Buildings, Machinery, &c., with fifty -six Steel Plates, 
containing over two hundred Diagrams. This work is designed to 
supply a want which the author experienced in his course of in- 
struction; and is the result of a well-digested plan which he found 
promotive of great benefit among his scholars. With a judgment 
that does not always accompany talent, the author has adapted his 
lessons to their practical application in every-day business, and thus 
while the scholar learns the art of Mechanical Drawing, he also 
learns what is equally essential, its adaptation to all useful pur- 
poses. We regard Mr. Minifie's work as one likely to confer 
great benefits on the rising generation, as a knowledge of what it 
teaches is of consequence to every one — to the mechanic who re- 
duces the art to practice, and to the merchant or the capitalist who 
tests the mechanic's skill by its application to his work." 



From the Baltimore Patriot, 

"The work must be highly appreciated by all whose studies or 
duties are directed to Drawing, and must, we should say, at once 
become a text book in all Scliools where this branch of learning is 
taught." 



From the Baltimore Siin. 

Drawing. — A Baltimore book of the first class. 
A text book designed unquestionably to take rank among the most 
valuable acquisitions of the art. The author exhibits in the details 
of the work the familiarity of a master with his subject; and, while 
with a dexterous hand simplifying his theme by the avoidance of 
technicalities as much as possible and the use of plain language, he 
adds to the attractiveness of the study, and lures the pupil by fa- 

2 



OPINIONS OF THE PRESS 



cilitatlng' the way to the highest attainments in the art. It will 
command beyond doubt an extensive popularity.'' 



From the Baltimore Western Continent. 

^'We are gratified to see a book of such sterling merit issued 
from a Baltimore publication house. All the externals are care- 
fully attended to. The engravings are all on steel, executed with 
great neatness and precision. The paper is white^ the type so 
clear that it is a real gratification to read it, and the binding solid 
and substantial. It may be thought strange that we mention such 
traits as these, but after reading books carelessly printed and so 
loosely bound that they fall to pieces on the first perusal, such ad- 
vantages as we have alluded to can be better appreciated. 

*'To come to the subject matter of the book. The lessons are 
(as they should be in a text book intended to be used not only by 
drawing masters, but also by those who wish to teach themselves 
the art of Mechanical Drawing,) progressive, beginning at the sim- 
plest rudiments, and gradually advancing to the higher and more 
complicated problems. During every stage of his advance, the 
student is furnished with numerous illustrations. He is required 
to do nothing of which a model is not supplied; and he is taught to 
study these models intelligently by a very clear and copious ex- 
planation of the necessary processes required, and a full definition 
of all the technical terms which are applicable to the figures pre- 
sented to him. Linear Drawing, the division of Angles, the for- 
mation of Polygons, the principles of Circles, the use of the va- 
rious Mathematical Instruments, the construction of the different 
Curvilinear figures take up the early lessons. The work then passes 
to the consideration of Solids and their Sections, more particularly 
those of the Cone and the Cylinder. This, of course, completes 
the study of the principles of the art. The student, if he have be- 
stowed on his lessons the necessary attention and care, has now 
mastered the alphabet of drawing. 

"At this point of the course comes in the practical application 
of the knowledge already acquired. The rules are first applied to. 
Building, to the laying of boards, the construction of arches, the 
drawing of plans and elevations of houses and the delineation of 
the various mouldings used in architecture. Machinery next takes 
its turn, then we have a treatise on Isometrical Drawing, prepara- 
tory to Perspective. The former is, in most works on this subject, 
included in the latter. 

"The pupil is now fully prepared to enter upon the study of 
Perspective. The principles of this important branch of drawing 
are laid down, and several examples given. We were pleased to 
find that Mr. Minifie has paid attention to the theory of Vanishing 
Points, and given rules for determining their position, a matter too 
often left by writers on Perspective to the blind arbitration of 



OPINIONS OP THE PRESS. 

chance. The theory of Shadows, which is properly a corrollary to 
Perspective^ closes the volume. 

^Mn taking leave of Mr. Minifie's book, we would heartily com- 
mend it to all persons who desire to acquire a knowledge of the 
important art of Mechanical Drawing. This forms a distinct and 
separate branch of art of the greatest practical importance to the 
builder, the machinist, and indeed to all artizans who work by 
plans laid down on a flat surface. The want of a work on this 
department of Graphics, thorough and at the same time easily in- 
telligible, has long been felt. This want we believe to be fully 
supplied by the volume before us, which contains all the rules ne- 
cessary to make a finished draughtsman of plans and models. He 
loho, having thoroughly mastered this book, cannot make any of 
the ordinary drawings of this kind, may well despair of ever being 
able io accomplish such a result.^ ^ 

From the Boston Post, 

"The title of this beautiful volume does not present a sufficiently 
elevated idea of either its character, usefulness, or outward ele- 
gance. It is, in the first place, one of the most handsomely print- 
ed works that we have ever seen, and in respect to illustrations is 
far superior to any scientific book which has ever come from the 
American press. It is fully equal to the most expensive of the 
English publications of its class. 

"It is intended both for Schools and individuals without a master, 
and it includes, therefore, the definitions and rules of Geometry fa- 
miliary explained, and very many of the practical problems, begin- 
nmg with the most simple, and embracing descriptions as void of 
technicalities as possible. The whole is illustrated by fifty-six 
Steel Plates, containing more than two hundred Diagrams. Plan 
drawing. Sections and Elevations of Buildings and Machinery are 
fully discussed and illustrated; and to these are added an introduc- 
tion to Isometrical Drawing and an essay on Linear Perspective 
and Shadows. The work is very intelligibly and systematically 
written, and we must repeat, that the illustrations are most beauti- 
fully engraved. A strong, substantial binding is a fitting finish to 
the internal excellence of the volume. We would cajl the at- 
tention of school committees and mechanics of all branches to the 
book under notice. It is evidently intended to be a standard one, 
and from what we know of the wants of both juvenile and adult 
students of Geometrical Drawing, is just the thing for both classes." 



From the Boston Journal. 

"The plan of this work is a good one — something of the kind 

is much wanted at this time — and so far as we are able to judge, 

the want is well supplied by the work before us, which is eminently 

practical in its character, and will be found useful to the Mechanic, 



OPINIONS OF THE PRESS. 



the Engineer, and the Architect. It is well adapted^ not only for 
private self-instruction, but as a text book of drawing to be used in 
our High Schools and Academies, where this useful branch of the 
line arts has been hitherto too much neglected." 



From the Boston CKronotype. 

'*An elementary practical text book on Perspective and Drawing 
as applied to objects of utility, such as Buildings, Machines, Sur- 
veys, &c. has been a great desideratum. All the works we have 
seen are either too superficial and elementary, or too scientific and 
extensive. Here is one adapted to meet the wants of a large class 
of learners and practical mechanics, which contains all the scientific 
principles that are ordinarily needed, with a plenty of practical 
problems. To be able to use mathematical instruments and cor- 
rectly to delineate any object, in true perspective, or as it actually 
appears, is a most useful and valuable faculty to any one, and con- 
sidering the ease with which the art may be acquired with proper 
teaching, it is wonderful that there is not more taught. It ought 
to be a part of every common school education. 

**A text book like this, which unites beauty, simplicity and sci- 
ence, bringing the most useful applications of the art of drawing 
within the reach of all, is truly a valuable gift to the cause of ed- 
ucation." 



From the Boston Atlas, 

^*Mintfie's Mechanical Drawing Book. — This is the title 
of a new work issued from the press of Wm. Minifie &. Co., Balti- 
more. It seems to be eminently well adapted as a text book, to 
the use of Schools, and students in Engineering, Architecture and 
Mechanics. It contains rules and illustrations for the application of 
the principles of Geometry to the drawing of Plans, Sections and 
elevations of Buildings and Machinery, the rules being simple and 
void of technicalities as much as is practicable. In the latter part 
of the work there are short treatises on Isometrical Drawing, Linear 
Perspective and Shadows, which will be found very useful. The 
work is illustrated with excellent Steel plates, containing over two 
hundred Diagrams. When it becomes known it will without 
doubt be extensively used in Schools and Colleges, as w^ell as 
for private self-instruction." 



Fiwn the Boston Daily Advertiser. 

^*The rules for drawing figures of every description on a plain 
surface, or in Perspective, by the simplest methods, are clearly 
given, based on mathematical principles. The definitions and de- 
scriptions have the merit of being both precise and intelligible, and 



OPINIONS OF THE PRESS. 



the work appears to be well adapted as a text book for High Schools, 
or a practical manual for architects.'^ 



From the Washington JYational Intelligencer. 

"A Text Book of Geometrical Drawing. — ^This is the 
title of a work recently published in Baltimore by an architect of 
established reputation there. The origin of it, Mr. Minifie ex- 
plains in his preface to have been in the necessities of the pupils to 
whom, as professor in the High School, he gives instruction and 
lectures on Mechanical Drawing and the applications of Geometry, 
and for whom no one existing text book furnished the solution of 
problems which are of daily practical occurrence, and which, for 
that very reason, are apt to be passed over as too simple for expo- 
sition by those who apply themselves to the more recondite topics 
of Mathematics. In this respect, Mr. Minifie has gone far to re- 
medy the deficiency; and while he does not assume to have origi- 
nated any thing new, he has presented, in a small and cheap com- 
pass and in a judicious and instructive arrangement, what otherwise 
had to be searched for in a dozen volumes, or more. For the young 
Mechanic, therefore, in furnishing the rules and proportions of lay- 
ing out and fitting work, this single book is a lihrary, 

^*Such completeness is not purchased, either by an undesirable 
or unintelligible brevity; but by a sedulous restriction on the part 
of the author to the aim with which he set out. This was the 
enabling of an ordinarily intelligent Mechanic to represent, on paper 
or on a board, either work that had heen, or work that had to he ex- 
ecuted in solid materials. Here, mere theory is unnecessary, fur- 
ther than to show the reasonableness of the several processes, and 
to assist the memory in retaining the sequence of the several steps; 
but distinct and literal rules, referred throughout to accompanying 
Diagrams, are copiously given for every thing, from the marking 
off of a mitre-joint to the development of the metal roofing of hemis- 
pherical domes, and from the erection of a perpendicular to the 
Perspective Drawing of extensive buildings. 

"'rhe style and order of arrangement seems to have arisen from 
a practical intimacy with the subject; those difficulties are chiefly 
dwelt on ivhich are likely to arise in actual working cases, and with 
actual workmen; and these are so treated, with minute and particu- 
lar rules, accompanied with abundant and perspicuous drawings, 
as to render the book quite adapted to the purpose of self instruction. 
Indeed, one who patiently and carefully goes through it could not 
be said, so far as principles and methods are concerned, any longer 
to want a master; while, as regards neatness of execution, the 
plates are quite sufficient to serve as models fur the learner. 

"On the whole, we have good authority for saying that so use- 
ful and so cheap a book for all who are concerned in the arts of 
building and design, has not appeared for a long time." 



OPINIONS OF THE PRESS. 



From the Washi7igton JYews. 



^^In every department of business or condition of life, drawing is 
not an accomplishment, but an indispensible necessity. With the 
inventive genius of our people, the art of drawing would be emi- 
nently useful, tending as it would, to advance the Mechanic and 
manufacturing arts. We esteem what is important for men to 
know as men, should be learned by children at school. To teach- 
ers we commend Minifie's Drawing Book; and to our hardy Me- 
chanics who have not time to devote to the study of long mathe- 
matical demonstrations, the book will be found eminently practical 
and useful. '^ 



From the JYeiv York Scientific American. 

*^It is the best work on Drawing that we have ever seen, and is 
especially a text book of Geometrical Drawing for the use of Me- 
chanics and Schools. No young mechanic, such as a Machinest, 
Engineer, Cabinet Maker, Millwright or Carpenter, 
should be without it. It is illustrated with fifty-six Steel Plates 
and contains more than two hundred Diagrams. The author, Mr. 
Minifie, shows that he is master of his subject in all its various 
branches, which he has illustrated with Plans, Sections, Eleva- 
tions, Perspective and Linear Views of Buildings and Machinery. 
Such Books — are Books. 

"The price is very moderate, considering the quality, the sterl- 
ing worth and style of the work.'' 



From the JS'exo York Farmer and MecJianic. 

"A Valuable Scientific Work. — We cannot forbear no- 
ticing more particularly an excellently arranged, highly practical 
and exceedingly valuable work recently published by Messrs. 
William Minifie & Co., at Baltimore, being a Text Book of 
Geometrical Drawing, and admirably calculated for the use of 
Mechanics, Schools and Academies. In this work the various rules 
and definitions of Geometry are familiarly explained with great 
distinctness, and the practical problems arranged, gradually pro- 
gressing from the most simple to the more complex, dispensing 
with the long and tedious detail of technicalities in description, so 
objectionable in many other works on the subject. 

'*The volume also contains some of the best illustrations for 
drawing Plans, Sections and Elevation of Machinery, and Build- 
ings that we have seen; including an introduction to Isometrical 
Drawing, and an essay on Linear Perspective and Shadows; and 
is beautifully illustrated with neariy sixty Steel Engravings. Mr. 
Minifie, the author, is an Architect of distinction, and a popular 
teacher of Drawing in the Central High School of Baltimore, and 
certainly in this work has given a proof of talents of a high order. 



OPINIONS OF THE PRESS. 

*^'The usefulness and capabilities of the application of the rules 
contained in this book, to the usual practical every-day business of 
life, is one of its best recommendations, and we believe its value 
will only require to be known, to cause it to be most extensively 
ad(^pted in Schools and Colleges as well as for private instruction. 
We most cheerfully commend it as a valuable acquisition to the 
libraries of all our mechanical readers.'' 



From the American Rail Road Journal. 

*'Minifie's Text Book of Mechanicl Drawing. — We are 

happy to call the attention of our readers to this work. It is re- 
commended by artists and gentlemen who are best qualified among 
us to judge correctly of its merits, as being the most thorough and 
complete loork of the kind ever published in this country, and as 
indispensable to those whose calling requires a knowledge of the 
principles of Geometrical Drawing or Sketching, and as a suitable 
guide and companion to amateurs in this delightful art. It has re- 
ceived universal commendation from the press, and we believe it 
fully merits all that has been said in its praise.'' 



From the JVew York Christian Ambassador, 

"We regard this work as invaluable. It ought to be in the 
hands of every mechanic. I am sorry to say, that many mechanics 
have no ambition for improvement in their business. They seem 
to be perfectly satisfied if they know how to do the manual part of 
their work, and make no effort to master the principles which they 
daily employ. A little study bestowed upon this excellent book, 
would give them a knowledge w^hich would be of vast service to 
them, and save them much hard labor. 

^*We would commend the work not only to mechanics, but also 
to School Committees. It ought to be introduced into our schools, 
that all the scholars who design to be mechanics, may be instructed 
in its principles." 



From the Philadelphia Public Ledger, 

"This will be found a most excellent text book, by an accom- 
plished teacher of Architectural and Mechanical Drawing. The 
problems are all selected with a view to their practical application 
in the every-day business of the Engineer, Architect and Artizan. 
It may be used as a self-instructor." 



From the Philadelphia JYorth American, 

^*A good and thorough text book, worthy the notice of teachers 
and young draughtsmen." 



OPINIONS OF THE PRESS. 

From McMakin's Model Journal, April 7. 

"It is^ indeed,, in all respects^ a very excellent and a very timely 
book.'^ 



From the Richmond Daily Whig. 

^'We regard it as a book evincing great thoroughness in the sub- 
ject on which it treats; one which must supply a deficiency which 
has long been felt, and one which must prove an invaluable assist- 
ant, either to the theoretical teacher or the practical artizan. It is 
a book also for self-instruction. I'he drawings are very numerous, 
and the plates admirably executed." 






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